{"id":71,"date":"2021-12-15T09:53:14","date_gmt":"2021-12-15T09:53:14","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/grenzwerte-von-funktionen\/"},"modified":"2021-12-15T09:53:14","modified_gmt":"2021-12-15T09:53:14","slug":"grenzwerte-von-funktionen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/grenzwerte-von-funktionen\/","title":{"raw":"Grenzwerte von Funktionen","rendered":"Grenzwerte von Funktionen"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: 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little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zfcf67adffbd5\" class=\"sectionHead\"><span class=\"titlemark\">6.4 <\/span> <a id=\"x1-1740004\"><\/a>Grenzwerte von Funktionen<\/h3> <p class=\"noindent\">Wir betrachten jetzt wieder allgemeine Funktionen <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> auf einer allgemeinen Teilmenge <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> und wollen (eigentliche und uneigentliche) Grenzwerte f\u00fcr den Fall definieren, wenn <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gegen ein <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> strebt (oder auch wenn <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> oder gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math> divergiert). <a id=\"x1-174001r173\"><\/a> <\/p> <h4 id=\"ze43694e12380\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.1 <\/span> <a id=\"x1-1750001\"><\/a>Grenzwerte und punktierte Umgebungen<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ein H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir erinnern daran, dass Letzteres genau dann der Fall ist, wenn <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-175001r7\" \/><mstyle class=\"maketag\"><mtext>(6.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> oder \u00e4quivalent, wenn es eine Folge in&nbsp;<math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> gibt, die gegen&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> strebt. <\/p><p class=\"indent\">F\u00fcr eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecbx-1095\">der Grenzwert<\/span> <span class=\"ecbx-1095\">von <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecbx-1095\">f<\/span><span class=\"ecbx-1095\">\u00fc<\/span><span class=\"ecbx-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> oder auch <span class=\"ecbx-1095\">der<\/span> <span class=\"ecbx-1095\">Grenzwert bei<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Informell ausgedr\u00fcckt bedeutet dies, dass die Funktionswerte von <math display=\"inline\"><mi>f<\/mi><\/math> beliebig nahe bei <math display=\"inline\"><mi>A<\/mi><\/math> liegen wenn&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> nahe an <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> heranr\u00fcckt. Der Grenzwert von <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> muss nat\u00fcrlich nicht existieren; wenn er existiert, ist er aber eindeutig bestimmt (diese Eigenschaft ist der Grund, wieso wir (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175001r7\">6.7<\/a>) angenommen haben, siehe \u00dcbung <a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175002r38\">6.38<\/a>). <\/p><p class=\"indent\">Der Grenzwert erf\u00fcllt, analog zu Proposition <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\">5.30<\/a>, die gewohnten Eigenschaften. Er ist <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">linear (das heisst, falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existieren, so existiert auch der Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und analog f\u00fcr skalare Multiplikation), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">multiplikativ (das heisst, falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existieren, so existiert auch <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math>), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">monoton (<math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>g<\/mi><\/math> impliziert <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> falls die Grenzwerte existieren) <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">und erf\u00fcllt ein Sandwich-Lemma.<\/div><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze3a830a6cf59\"> <a id=\"x1-175002r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.38 <\/span>(Erste Eigenschaften)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Beweisen Sie, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eindeutig bestimmt ist, falls er existiert.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Beweisen Sie die drei Eigenschaften linear, multiplikativ und monoton des Grenzwerts<\/span> <span class=\"ecti-1095\">von Funktionen auf <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren und beweisen Sie ein Sandwich-Lemma f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Grenzwert von Funktionen<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z1f5547b1e171\"> <a id=\"x1-175006r39\"><\/a> <span class=\"ecbx-1095\">Lemma 6.39 <\/span>(Grenzwerte und Stetigkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge, <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eine reellwertige Funktion auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">genau dann stetig bei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wenn<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Falls&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> stetig ist, dann existiert zu jedem&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math>                                                                                                                                                                           ein&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> die Implikation <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> gilt. Vergleicht man dies mit der Definition von&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> erh\u00e4lt man&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Falls umgekehrt&nbsp;<math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> gilt, so m\u00fcssen wir wiederum nur die Definition der Stetigkeit (und die Gleichheit des Grenzwerts mit&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math>) verwenden, um Stetigkeit von&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> zu erhalten. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Obiges Lemma hat auch eine Interpretation f\u00fcr den Fall <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2209<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> denn in diesem Fall w\u00e4re der Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> (falls dieser existiert) ein guter Kandidat f\u00fcr eine Fortsetzung der Funktion auf die Menge <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> da diese Fortsetzung dann bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> stetig wird. <\/p><p class=\"indent\">Man nennt einen H\u00e4ufungspunkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> eine <span class=\"ecbx-1095\">hebbare Unstetigkeitsstelle <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existiert, aber nicht gleich <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist (siehe auch Figur&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-177001r5\">6.5<\/a>). In diesem Fall kann man eine neue Funktion <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mstyle class=\"text\"><mtext>neu<\/mtext><\/mstyle> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mstyle class=\"text\"><mtext>neu<\/mtext><\/mstyle><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> definieren, die bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> stetig ist.                                                                                                                                                                           <\/p><p class=\"indent\">Genauso wie in dem Beweis von Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> sieht man nun, dass f\u00fcr eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math> mit <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> die Gleichheit <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> gilt, falls letzter Grenzwert existiert. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"zaf058ec520a5\"> <a id=\"x1-175007r40\"><\/a> <span class=\"ecbx-1095\">Lemma 6.40 <\/span>(Grenzwerte mittels Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">genau dann, wenn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> ist eine Folge in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Dann existiert f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren existiert ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was gemeinsam <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ergibt. Die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> konvergiert also gegen <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">F\u00fcr die Umkehrung nehmen wir an, dass&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> nicht erf\u00fcllt ist (also entweder der Grenzwert nicht existiert oder nicht gleich&nbsp;<math display=\"inline\"><mi>A<\/mi><\/math> ist). Dann existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           so dass f\u00fcr alle <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> existiert mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi> <mo class=\"MathClass-bin\">\u2227<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir verwenden dies f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und finden also ein <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-175008r8\" \/><mstyle class=\"maketag\"><mtext>(6.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-175009r9\" \/><mstyle class=\"maketag\"><mtext>(6.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Aus Ungleichung (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175008r8\">6.8<\/a>) schliessen wir, dass die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> Werte in&nbsp;<math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math> annimmt und gegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> konvergiert. Aus Ungleichung (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175009r9\">6.9<\/a>) folgt, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nicht gegen <math display=\"inline\"><mi>A<\/mi><\/math> konvergiert. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z7e993cb23e5e\"> <a id=\"x1-175010r41\"><\/a> <span class=\"ecbx-1095\">Proposition 6.41 <\/span>(Grenzwerte und Verkn\u00fcpfung mit stetigen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion,<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>E<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine bei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">stetige Funktion. Dann gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">\ud83e\ude86Dies ist ein Matrjoschka-Beweis. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine Folge in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> mit&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Nach Lemma&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a> gilt dann <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Die Stetigkeit von&nbsp;<math display=\"inline\"><mi>g<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> impliziert nun gemeinsam mit Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>, dass&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Da&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine beliebige Folge in&nbsp;<math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math>                                                                                                                                                                           mit&nbsp;<math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> war, folgt wiederum aus Lemma&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a>, dass&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Diese Eigenschaften von Grenzwerten k\u00f6nnen bereits f\u00fcr die Berechnung von vielen Grenzwerten verwendet werden. <\/p><p class=\"indent\">Weiters k\u00f6nnen wir uneigentliche Grenzwerte definieren. Wir sagen zum Beispiel, dass&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gegen&nbsp;<math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> divergiert und schreiben&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir nennen die Menge&nbsp;<math display=\"inline\"><msub><mrow><mover accent=\"true\"><mrow><mi>U<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die <span class=\"ecbx-1095\">punktierte<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecbx-1095\">-Umgebung<\/span> um <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> und bemerken, dass diese Mengen implizit in der Definition des Grenzwerts aufgetreten sind. <a id=\"x1-175011r175\"><\/a> <\/p> <h4 id=\"za352e2d55cfa\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.2 <\/span> <a id=\"x1-1760002\"><\/a>Links- und rechtsseitige Grenzwerte<\/h4> <p class=\"noindent\">Angenommen <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist eine Teilmenge und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> hat die Eigenschaft <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Intuitiv hat der Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> also die Eigenschaft, dass ihm <math display=\"inline\"><mi>D<\/mi><\/math> von rechts beliebig nahe kommt, was also eine st\u00e4rkere Forderung ist als (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175001r7\">6.7<\/a>). Einen solchen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> wollen wir einen <span class=\"ecbx-1095\">rechtsseitigen H<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ufungspunkt <\/span>von                                                                                                                                                                           <math display=\"inline\"><mi>D<\/mi><\/math> nennen. F\u00fcr eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> (alternativ <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msubsup><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> oder auch <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">&gt;<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math>) der <span class=\"ecbx-1095\">rechtsseitige<\/span> <span class=\"ecbx-1095\">Grenzwert von <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecbx-1095\">bei <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir schreiben <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> die Eigenschaft <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/math> f\u00fcr alle&nbsp;<math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> hat (<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ist ein <span class=\"ecbx-1095\">linksseitiger H<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ufungspunkt<\/span>), k\u00f6nnen wir ebenso den <span class=\"ecbx-1095\">linksseitigen Grenzwert<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> (alternativ <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msubsup><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> oder auch <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">&lt;<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math>) definieren. <\/p><p class=\"indent\">Falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein links- und rechtsseitiger H\u00e4ufungspunkt ist, dann existiert der Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> genau dann, wenn die links- und rechtseitigen Grenzwerte von <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> existiert und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> erf\u00fcllt ist. <\/p><p class=\"indent\">Beispiele von links- und rechtsseitigen Grenzwerten sind <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <a id=\"x1-176001r176\"><\/a> <h4 id=\"z588a09e231e0\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.3 <\/span> <a id=\"x1-1770003\"><\/a>Einseitige Stetigkeit und Sprungstellen<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein rechtseitiger                                                                                                                                                                           H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <mi>D<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/> <mo class=\"MathClass-rel\">\u2192<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u211d<\/mi><\/math> eine Funktion. Falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existiert und gleich <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> ist, dann sagen wir, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">rechtsseitig<\/span> <span class=\"ecbx-1095\">stetig <\/span>bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ist. Ist <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein linksseitiger H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> dann sagen wir analog, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">linksseitig stetig <\/span>bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ist, falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existiert und gleich <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=598&amp;url=Pictures\/folgen\/lstetig\/lstetigtot.pdf\"><img id=\"zb2d4f14f2a5d\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/folgen\/lstetig\/lstetigtot.svg\" width=\"598\"><\/div> <a id=\"x1-177001r5\"><\/a> <a id=\"x1-177002\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;6.5:               <\/span><span class=\"content\">Der  Graph  einer  Funktion,  die  eine  hebbare  Unstetigkeitsstelle  bei               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math>                   hat,                                                                                     bei               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math>                   linksseitig   stetig   (aber   nicht   rechtseitig   stetig)   ist   und   bei               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math>                   rechtsseitig stetig ist.                                                                  &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">Sei nun <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein links- und rechtsseitiger H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> (insbesondere ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math>). F\u00fcr eine Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> heisst&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> eine <span class=\"ecbx-1095\">Sprungstelle<\/span>, falls die einseitigen Grenzwerte&nbsp;<math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existieren, aber verschieden sind. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zdad79605ca11\"> <a id=\"x1-177003r42\"><\/a> <span class=\"ecbx-1095\">Applet 6.42 <\/span>(Grenzwerte einer Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"400px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/vzk4xHNv\/width\/688\/height\/400\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen eine Funktion mit Definitionsbereich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>8<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>8<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und betrachten  verschiedene  Bewegungen  im  Definitionsbereich  und  Grenzwerte  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  diese<\/span> <span class=\"ecti-1095\">Funktion.<\/span> <\/p> <\/div> <a id=\"x1-177004r177\"><\/a> <h4 id=\"z7aa1aeec290d\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.4 <\/span> <a id=\"x1-1780004\"><\/a>Die Bewegung nach Unendlich<\/h4> <p class=\"noindent\">Angenommen <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist eine nicht von oben beschr\u00e4nkte Teilmenge (das heisst, f\u00fcr alle <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gilt <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/math>) und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist eine Funktion. Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> gegen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> strebt f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> und schreiben <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfe08838428b2\"> <a id=\"x1-178001r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.43 <\/span>(Beispiele f\u00fcr uneigentliche Grenzwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Definieren Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">wie oben und eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die uneigentlichen Grenzwerte <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">und finden Sie je eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-178002r178\"><\/a> <h4 id=\"ze2e128270bc4\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.5 <\/span> <a id=\"x1-1790005\"><\/a>Einige Rechenbeispiele<\/h4> <p class=\"noindent\">Wir werden bei Rechnungen wie den folgenden oft davon ausgehen, dass ein Formelausdruck eine Funktion mit dem maximalen f\u00fcr den Formelausdruck sinnvollen Definitionsbereich definiert. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2feb6110d57a\"> <a id=\"x1-179001r44\"><\/a> <span class=\"ecbx-1095\">Beispiel 6.44.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen hier<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">berechnen, und m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r dies zwei weitere Grenzwerte berechnen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Wir behaupten zuerst<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>y<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>y<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-179002r10\" \/><mstyle class=\"maketag\"><mtext>(6.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">In der Tat gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">auf Grund der<\/span> <span class=\"ecti-1095\">Monotonie der Folge<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die in Abschnitt <\/span><a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1650003\"><span class=\"ecti-1095\">6.3<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Definition der Exponentialabbildung verwendet wurde. Daraus ergibt<\/span> <span class=\"ecti-1095\">sich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>4<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was wegen dem Sandwich-Lemma (Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"#x1-303006r6\"><span class=\"ecti-1095\">B.6<\/span><\/a><span class=\"ecti-1095\">) eben<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179002r10\">6.10<\/a>) <span class=\"ecti-1095\">impliziert.<\/span> <\/p><\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Als n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chstes wollen wir<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-179003r11\" \/><mstyle class=\"maketag\"><mtext>(6.11)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">zeigen. Sei also<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">gibt es wegen<\/span> (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179002r10\">6.10<\/a>) <span class=\"ecti-1095\">ein<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann ist<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">auf Grund der strengen Monotonie der Logarithmus-Abbildung und<\/span> <span class=\"ecti-1095\">damit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was zu zeigen war.<\/span> <\/p><\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Auf Grund von Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175010r41\"><span class=\"ecti-1095\">6.41<\/span><\/a> <span class=\"ecti-1095\">und da die Exponentialabbildung stetig ist, ergibt sich aus<\/span> (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179003r11\">6.11<\/a>)<span class=\"ecti-1095\">nun<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Hieraus ergibt sich auch ein weiterer Beweis f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Beispiel <\/span><a href=\"..\/..\/chapter\/reelle-folgen#x1-157006r4\"><span class=\"ecti-1095\">6.4<\/span><\/a> <span class=\"ecti-1095\">(iii). (Wieso?<\/span><button class=\"hover-trigger\"><span class=\"ecti-1095\">(Wieso?)<\/span><\/button><span class=\"hover-text\"><span class=\"marginpar\"><span class=\"ecti-1095\">Nach Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\"><span class=\"ecti-1095\">6.40<\/span><\/a> <span class=\"ecti-1095\">folgt<\/span> <span class=\"ecti-1095\">aus obigem <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nimmt man nun den Kehrwert so erhalten wir mit Proposition <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a> <span class=\"ecti-1095\">(iii) den Grenzwert<\/span> <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>n<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/span><\/span><span class=\"ecti-1095\">)<\/span><\/p><\/div><\/div> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z80edfa6635ca\"> <a id=\"x1-179004r45\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.45.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die folgenden Grenzwerte (falls sie existieren)<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>2<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>3<\/mn><msup><mrow><mi>e<\/mi><\/mrow><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>e<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>e<\/mi><\/mrow><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>e<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Beschreiben Sie weiters, wie man die Grenzwerte<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>p<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r zwei Polynome<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">berechnet.<\/span> <\/p> <\/div> <a id=\"x1-179005r174\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: 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*\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zfcf67adffbd5\" class=\"sectionHead\"><span class=\"titlemark\">6.4 <\/span> <a id=\"x1-1740004\"><\/a>Grenzwerte von Funktionen<\/h3> <p class=\"noindent\">Wir betrachten jetzt wieder allgemeine Funktionen <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> auf einer allgemeinen Teilmenge <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> und wollen (eigentliche und uneigentliche) Grenzwerte f\u00fcr den Fall definieren, wenn <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gegen ein <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> strebt (oder auch wenn <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> oder gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math> divergiert). <a id=\"x1-174001r173\"><\/a> <\/p> <h4 id=\"ze43694e12380\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.1 <\/span> <a id=\"x1-1750001\"><\/a>Grenzwerte und punktierte Umgebungen<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ein H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir erinnern daran, dass Letzteres genau dann der Fall ist, wenn <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-175001r7\" \/><mstyle class=\"maketag\"><mtext>(6.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> oder \u00e4quivalent, wenn es eine Folge in&nbsp;<math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> gibt, die gegen&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> strebt. <\/p><p class=\"indent\">F\u00fcr eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecbx-1095\">der Grenzwert<\/span> <span class=\"ecbx-1095\">von <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecbx-1095\">f<\/span><span class=\"ecbx-1095\">\u00fc<\/span><span class=\"ecbx-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> oder auch <span class=\"ecbx-1095\">der<\/span> <span class=\"ecbx-1095\">Grenzwert bei<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Informell ausgedr\u00fcckt bedeutet dies, dass die Funktionswerte von <math display=\"inline\"><mi>f<\/mi><\/math> beliebig nahe bei <math display=\"inline\"><mi>A<\/mi><\/math> liegen wenn&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> nahe an <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> heranr\u00fcckt. Der Grenzwert von <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> muss nat\u00fcrlich nicht existieren; wenn er existiert, ist er aber eindeutig bestimmt (diese Eigenschaft ist der Grund, wieso wir (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175001r7\">6.7<\/a>) angenommen haben, siehe \u00dcbung <a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175002r38\">6.38<\/a>). <\/p><p class=\"indent\">Der Grenzwert erf\u00fcllt, analog zu Proposition <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\">5.30<\/a>, die gewohnten Eigenschaften. Er ist <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">linear (das heisst, falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existieren, so existiert auch der Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und analog f\u00fcr skalare Multiplikation), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">multiplikativ (das heisst, falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existieren, so existiert auch <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math>), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">monoton (<math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>g<\/mi><\/math> impliziert <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> falls die Grenzwerte existieren) <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">und erf\u00fcllt ein Sandwich-Lemma.<\/div><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze3a830a6cf59\"> <a id=\"x1-175002r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.38 <\/span>(Erste Eigenschaften)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Beweisen Sie, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eindeutig bestimmt ist, falls er existiert.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Beweisen Sie die drei Eigenschaften linear, multiplikativ und monoton des Grenzwerts<\/span> <span class=\"ecti-1095\">von Funktionen auf <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren und beweisen Sie ein Sandwich-Lemma f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Grenzwert von Funktionen<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1f5547b1e171\"> <a id=\"x1-175006r39\"><\/a> <span class=\"ecbx-1095\">Lemma 6.39 <\/span>(Grenzwerte und Stetigkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge, <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eine reellwertige Funktion auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">genau dann stetig bei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wenn<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Falls&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> stetig ist, dann existiert zu jedem&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math>                                                                                                                                                                           ein&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> die Implikation <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> gilt. Vergleicht man dies mit der Definition von&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> erh\u00e4lt man&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Falls umgekehrt&nbsp;<math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> gilt, so m\u00fcssen wir wiederum nur die Definition der Stetigkeit (und die Gleichheit des Grenzwerts mit&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math>) verwenden, um Stetigkeit von&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> zu erhalten. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Obiges Lemma hat auch eine Interpretation f\u00fcr den Fall <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2209<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> denn in diesem Fall w\u00e4re der Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> (falls dieser existiert) ein guter Kandidat f\u00fcr eine Fortsetzung der Funktion auf die Menge <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> da diese Fortsetzung dann bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> stetig wird. <\/p><p class=\"indent\">Man nennt einen H\u00e4ufungspunkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> eine <span class=\"ecbx-1095\">hebbare Unstetigkeitsstelle <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existiert, aber nicht gleich <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist (siehe auch Figur&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-177001r5\">6.5<\/a>). In diesem Fall kann man eine neue Funktion <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mstyle class=\"text\"><mtext>neu<\/mtext><\/mstyle> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mstyle class=\"text\"><mtext>neu<\/mtext><\/mstyle><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> definieren, die bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> stetig ist.                                                                                                                                                                           <\/p><p class=\"indent\">Genauso wie in dem Beweis von Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> sieht man nun, dass f\u00fcr eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math> mit <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> die Gleichheit <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> gilt, falls letzter Grenzwert existiert. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zaf058ec520a5\"> <a id=\"x1-175007r40\"><\/a> <span class=\"ecbx-1095\">Lemma 6.40 <\/span>(Grenzwerte mittels Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">genau dann, wenn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> ist eine Folge in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Dann existiert f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren existiert ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was gemeinsam <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ergibt. Die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> konvergiert also gegen <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">F\u00fcr die Umkehrung nehmen wir an, dass&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> nicht erf\u00fcllt ist (also entweder der Grenzwert nicht existiert oder nicht gleich&nbsp;<math display=\"inline\"><mi>A<\/mi><\/math> ist). Dann existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           so dass f\u00fcr alle <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> existiert mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi> <mo class=\"MathClass-bin\">\u2227<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir verwenden dies f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und finden also ein <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-175008r8\" \/><mstyle class=\"maketag\"><mtext>(6.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-175009r9\" \/><mstyle class=\"maketag\"><mtext>(6.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Aus Ungleichung (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175008r8\">6.8<\/a>) schliessen wir, dass die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> Werte in&nbsp;<math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math> annimmt und gegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> konvergiert. Aus Ungleichung (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175009r9\">6.9<\/a>) folgt, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nicht gegen <math display=\"inline\"><mi>A<\/mi><\/math> konvergiert. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7e993cb23e5e\"> <a id=\"x1-175010r41\"><\/a> <span class=\"ecbx-1095\">Proposition 6.41 <\/span>(Grenzwerte und Verkn\u00fcpfung mit stetigen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion,<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>E<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine bei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">stetige Funktion. Dann gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">\ud83e\ude86Dies ist ein Matrjoschka-Beweis. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine Folge in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> mit&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Nach Lemma&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a> gilt dann <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Die Stetigkeit von&nbsp;<math display=\"inline\"><mi>g<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> impliziert nun gemeinsam mit Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>, dass&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Da&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine beliebige Folge in&nbsp;<math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math>                                                                                                                                                                           mit&nbsp;<math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> war, folgt wiederum aus Lemma&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a>, dass&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Diese Eigenschaften von Grenzwerten k\u00f6nnen bereits f\u00fcr die Berechnung von vielen Grenzwerten verwendet werden. <\/p><p class=\"indent\">Weiters k\u00f6nnen wir uneigentliche Grenzwerte definieren. Wir sagen zum Beispiel, dass&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gegen&nbsp;<math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> divergiert und schreiben&nbsp;<span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir nennen die Menge&nbsp;<math display=\"inline\"><msub><mrow><mover accent=\"true\"><mrow><mi>U<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die <span class=\"ecbx-1095\">punktierte<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecbx-1095\">-Umgebung<\/span> um <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> und bemerken, dass diese Mengen implizit in der Definition des Grenzwerts aufgetreten sind. <a id=\"x1-175011r175\"><\/a> <\/p> <h4 id=\"za352e2d55cfa\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.2 <\/span> <a id=\"x1-1760002\"><\/a>Links- und rechtsseitige Grenzwerte<\/h4> <p class=\"noindent\">Angenommen <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist eine Teilmenge und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> hat die Eigenschaft <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Intuitiv hat der Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> also die Eigenschaft, dass ihm <math display=\"inline\"><mi>D<\/mi><\/math> von rechts beliebig nahe kommt, was also eine st\u00e4rkere Forderung ist als (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175001r7\">6.7<\/a>). Einen solchen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> wollen wir einen <span class=\"ecbx-1095\">rechtsseitigen H<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ufungspunkt <\/span>von                                                                                                                                                                           <math display=\"inline\"><mi>D<\/mi><\/math> nennen. F\u00fcr eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> (alternativ <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msubsup><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> oder auch <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">&gt;<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math>) der <span class=\"ecbx-1095\">rechtsseitige<\/span> <span class=\"ecbx-1095\">Grenzwert von <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecbx-1095\">bei <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir schreiben <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> die Eigenschaft <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/math> f\u00fcr alle&nbsp;<math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> hat (<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ist ein <span class=\"ecbx-1095\">linksseitiger H<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ufungspunkt<\/span>), k\u00f6nnen wir ebenso den <span class=\"ecbx-1095\">linksseitigen Grenzwert<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> (alternativ <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msubsup><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> oder auch <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">&lt;<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math>) definieren. <\/p><p class=\"indent\">Falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein links- und rechtsseitiger H\u00e4ufungspunkt ist, dann existiert der Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> genau dann, wenn die links- und rechtseitigen Grenzwerte von <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> existiert und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> erf\u00fcllt ist. <\/p><p class=\"indent\">Beispiele von links- und rechtsseitigen Grenzwerten sind <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>0<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <a id=\"x1-176001r176\"><\/a> <h4 id=\"z588a09e231e0\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.3 <\/span> <a id=\"x1-1770003\"><\/a>Einseitige Stetigkeit und Sprungstellen<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein rechtseitiger                                                                                                                                                                           H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <mi>D<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/> <mo class=\"MathClass-rel\">\u2192<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u211d<\/mi><\/math> eine Funktion. Falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existiert und gleich <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> ist, dann sagen wir, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">rechtsseitig<\/span> <span class=\"ecbx-1095\">stetig <\/span>bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ist. Ist <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein linksseitiger H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> dann sagen wir analog, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">linksseitig stetig <\/span>bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ist, falls <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existiert und gleich <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=598&amp;url=Pictures\/folgen\/lstetig\/lstetigtot.pdf\"><img decoding=\"async\" id=\"zb2d4f14f2a5d\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/folgen\/lstetig\/lstetigtot.svg\" width=\"598\" \/><\/div> <a id=\"x1-177001r5\"><\/a> <a id=\"x1-177002\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;6.5:               <\/span><span class=\"content\">Der  Graph  einer  Funktion,  die  eine  hebbare  Unstetigkeitsstelle  bei               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math>                   hat,                                                                                     bei               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math>                   linksseitig   stetig   (aber   nicht   rechtseitig   stetig)   ist   und   bei               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math>                   rechtsseitig stetig ist.                                                                  &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">Sei nun <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein links- und rechtsseitiger H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> (insbesondere ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math>). F\u00fcr eine Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> heisst&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> eine <span class=\"ecbx-1095\">Sprungstelle<\/span>, falls die einseitigen Grenzwerte&nbsp;<math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> existieren, aber verschieden sind. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zdad79605ca11\"> <a id=\"x1-177003r42\"><\/a> <span class=\"ecbx-1095\">Applet 6.42 <\/span>(Grenzwerte einer Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"400px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/vzk4xHNv\/width\/688\/height\/400\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen eine Funktion mit Definitionsbereich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>8<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>8<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und betrachten  verschiedene  Bewegungen  im  Definitionsbereich  und  Grenzwerte  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  diese<\/span> <span class=\"ecti-1095\">Funktion.<\/span> <\/p> <\/div> <a id=\"x1-177004r177\"><\/a> <h4 id=\"z7aa1aeec290d\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.4 <\/span> <a id=\"x1-1780004\"><\/a>Die Bewegung nach Unendlich<\/h4> <p class=\"noindent\">Angenommen <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist eine nicht von oben beschr\u00e4nkte Teilmenge (das heisst, f\u00fcr alle <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gilt <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/math>) und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist eine Funktion. Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> gegen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> strebt f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> und schreiben <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfe08838428b2\"> <a id=\"x1-178001r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.43 <\/span>(Beispiele f\u00fcr uneigentliche Grenzwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Definieren Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">wie oben und eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die uneigentlichen Grenzwerte <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">und finden Sie je eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-178002r178\"><\/a> <h4 id=\"ze2e128270bc4\" class=\"subsectionHead\"><span class=\"titlemark\">6.4.5 <\/span> <a id=\"x1-1790005\"><\/a>Einige Rechenbeispiele<\/h4> <p class=\"noindent\">Wir werden bei Rechnungen wie den folgenden oft davon ausgehen, dass ein Formelausdruck eine Funktion mit dem maximalen f\u00fcr den Formelausdruck sinnvollen Definitionsbereich definiert. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2feb6110d57a\"> <a id=\"x1-179001r44\"><\/a> <span class=\"ecbx-1095\">Beispiel 6.44.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen hier<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">berechnen, und m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r dies zwei weitere Grenzwerte berechnen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Wir behaupten zuerst<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>y<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>y<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-179002r10\" \/><mstyle class=\"maketag\"><mtext>(6.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">In der Tat gilt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">auf Grund der<\/span> <span class=\"ecti-1095\">Monotonie der Folge<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die in Abschnitt <\/span><a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1650003\"><span class=\"ecti-1095\">6.3<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Definition der Exponentialabbildung verwendet wurde. Daraus ergibt<\/span> <span class=\"ecti-1095\">sich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>4<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was wegen dem Sandwich-Lemma (Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"#x1-303006r6\"><span class=\"ecti-1095\">B.6<\/span><\/a><span class=\"ecti-1095\">) eben<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179002r10\">6.10<\/a>) <span class=\"ecti-1095\">impliziert.<\/span> <\/p><\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Als n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chstes wollen wir<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-179003r11\" \/><mstyle class=\"maketag\"><mtext>(6.11)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">zeigen. Sei also<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">gibt es wegen<\/span> (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179002r10\">6.10<\/a>) <span class=\"ecti-1095\">ein<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann ist<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">auf Grund der strengen Monotonie der Logarithmus-Abbildung und<\/span> <span class=\"ecti-1095\">damit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was zu zeigen war.<\/span> <\/p><\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Auf Grund von Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175010r41\"><span class=\"ecti-1095\">6.41<\/span><\/a> <span class=\"ecti-1095\">und da die Exponentialabbildung stetig ist, ergibt sich aus<\/span> (<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-179003r11\">6.11<\/a>)<span class=\"ecti-1095\">nun<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Hieraus ergibt sich auch ein weiterer Beweis f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Beispiel <\/span><a href=\"..\/..\/chapter\/reelle-folgen#x1-157006r4\"><span class=\"ecti-1095\">6.4<\/span><\/a> <span class=\"ecti-1095\">(iii). (Wieso?<\/span><button class=\"hover-trigger\"><span class=\"ecti-1095\">(Wieso?)<\/span><\/button><span class=\"hover-text\"><span class=\"marginpar\"><span class=\"ecti-1095\">Nach Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\"><span class=\"ecti-1095\">6.40<\/span><\/a> <span class=\"ecti-1095\">folgt<\/span> <span class=\"ecti-1095\">aus obigem <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nimmt man nun den Kehrwert so erhalten wir mit Proposition <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a> <span class=\"ecti-1095\">(iii) den Grenzwert<\/span> <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>n<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/span><\/span><span class=\"ecti-1095\">)<\/span><\/p><\/div><\/div> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z80edfa6635ca\"> <a id=\"x1-179004r45\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.45.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die folgenden Grenzwerte (falls sie existieren)<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>2<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>3<\/mn><msup><mrow><mi>e<\/mi><\/mrow><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>e<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>e<\/mi><\/mrow><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>e<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Beschreiben Sie weiters, wie man die Grenzwerte<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>p<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r zwei Polynome<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">berechnet.<\/span> <\/p> <\/div> <a id=\"x1-179005r174\"><\/a> 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