Bis 2.5 Grenzwerte

Band2/B2.1.tex

@@ -1,74 +0,0 @@
-%!tikz editor 1.0
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-
-\addplot[domain=-2:2, blue, line width=1,samples=2000] {x^2};
-
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-
-
-
-\scriptsize{ \node () at (axis cs:2.4,1.8) {$\displaystyle y=\left\{ \begin{array}{rl}
-\frac{x^2-\frac{1}{4}}{x-\frac{1}{2}} & \text{für}\; x \neq \frac{1}{2}\\ \\
-2 & \text{für}\; x = \frac{1}{2}\\
-\end{array}
-\right .$};}
-
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-% \node () at (axis cs:1.3,-0.25) {$x_0$};
-
-% \draw[] (axis cs:1.14,-0.05) -- (axis cs:1.14, 0.05);
-% \node () at (axis cs:0.33,-0.45) {$\frac{1}{\pi}$};
-
-\end{axis}
-\end{tikzpicture}
-%!tikz source end
-
-\end{document}

Band2/B2.10.tikz

@@ -1,37 +0,0 @@
-\begin{tikzpicture}
-
-\begin{axis}[
- x=1cm,y=1cm,
-axis lines=middle,
-      axis x line=middle,
-        axis y line=middle,
-        %enlarge x limits=0.15,
-        %enlarge y limits=0.15,
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-        every axis y label/.style={at={(current axis.above origin)},anchor=north east},
-xmin=-0.5,
-xmax=6.,
-ymin=-0.5,
-ymax=5,
-ytick={1,...,2},
-xtick={3},
-xlabel=$x$,
-ylabel=$y$,
-]
-
-\addplot[domain=0.75:3, blue!80!black, line width=1,samples=50] {3/x};
-\addplot[domain=3:5, blue!80!black, line width=1,samples=50] {x-1};
-\addplot[color=blue!80!black, only marks, style={mark=*, fill=white}] coordinates {(3,2)};
-\addplot[color=blue!80!black, only marks, style={mark=*	}] coordinates {(3,1)};
-
-\draw [dashed, draw=black] (axis cs: -0.05,1) -- (axis cs: 3,1);
-\draw [dashed, draw=black] (axis cs: -0.05,2) -- (axis cs: 3,2);
-\draw [dashed, draw=black] (axis cs: 3,-0.05) -- (axis cs: 3,2);
-
-\node[] at (axis cs:4.5,4.5) {\footnotesize$y=\left\{\begin{array}{l}
-\frac{3}{x} \text { für } 0<x \leq 3 \\
-x-1 \text { für } x>3
-\end{array}\right.$};
-
-\end{axis}
-\end{tikzpicture}

Band2/B2.11.tikz

@@ -1,33 +0,0 @@
-\begin{tikzpicture}[line cap=round,line join=round,x=1cm,y=1cm]
-\tikzset{
-        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
-        small dot/.style={fill=black,circle,scale=0.3},}
-\begin{axis}[
-x=1.5cm,y=1.5cm,
-axis lines=middle,
-axis line style = {-latex},
-xmin=-0.25,
-xmax=4.2,
-ymin=-0.25,
-ymax=3,
-ytick=\empty,
-xtick=\empty,
-xlabel=$x$,
-ylabel=$y$,
-extra y ticks={1.5},
- extra y tick style={
-        tick label style={anchor=east}},
-    extra y tick labels={$\displaystyle g$},
-enlargelimits = true,
-]
-\draw [thick, draw=blue!50!black] 
-        (axis cs: -0.05,1.5) -- (axis cs: 5.2,1.5);
-\node[] at (axis cs:2.5,2.5) {$y=f(x)$};
-
-\addplot[domain=1.1:4.4, green!50!black	, line width=1,samples=200] {1.5+(5*e^(-x))*(cos(deg(2*pi*x)))};
-
-\end{axis}
-
-
-
-\end{tikzpicture}

Band2/B2.12.tikz

@@ -1,24 +0,0 @@
-\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm]
-\begin{axis}[
-x=1cm,y=1cm,
-axis lines=middle,
-xmin=-5,
-xmax=5,
-ymin=-1.4,
-ymax=5,
-ytick={-1,...,4},
-xtick={-3,...,3},
-xlabel=$x$,
-ylabel=$y$,
-]
-
-
-\addplot[domain=0.5:4, blue, line width=1,samples=50] {1/(x^2)};
-\addplot[domain=-4:-0.5, blue, line width=1,samples=50] {1/(x^2)};
-
-\node[] at (axis cs:2.5,2.5) {$\displaystyle y=\frac{1}{x^2}\; (x \neq 0)$};
-\end{axis}
-
-
-
-\end{tikzpicture}

Band2/B2.3.tikz

@@ -1,27 +0,0 @@
-\begin{tikzpicture}[line cap=round,line join=round,x=1cm,y=1cm]
-\tikzset{
-        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
-        small dot/.style={fill=black,circle,scale=0.5},}
-\begin{axis}[
-        axis x line=middle,
-        x=3cm,
-        width=40mm,
-        height=20mm,
-        xmin=0.5,
-        xmax=3.5,
-        xtick = \empty,
-        x axis line style={thick,-latex},
-        xlabel style={anchor=north west},
-        axis y line=none,
-        anchor=left of origin,
-xlabel=$x$,
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-extra x tick labels={$x_0 - C$, $x_0$, $x_0+C$},
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-\addplot[only marks, fill=white, thick] coordinates {(2,0)};
-\addplot[only marks, fill=white, thick] coordinates {(3,0)};
-\end{axis}
-
-\end{tikzpicture}

Band2/B2.4.tikz

@@ -1,53 +0,0 @@
-\begin{tikzpicture}
-[line cap=round,line join=round,x=1cm,y=1cm]
-\tikzset{
-        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
-        small dot/.style={fill=black,circle,scale=0.3},}
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-axis lines=middle,
-axis line style = {-latex},
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-extra y ticks={0.64, 1.5, 2.0625, 2.46},
-extra y tick labels={$f(x_1)$, $f(x_3)$, $g$ , $f(x_2)$},
-]
-
-\addplot[domain=0.5:3.1, blue, line width=1,samples=500] {0.25*(x^2)+0.5};
-
-
-\draw [line width=0.1] (axis cs: 0.75,-0.05) -- (axis cs: 0.75,0.640625);
-\draw [line width=0.1] (axis cs: -0.05,0.640625) -- (axis cs: 0.75,0.640625);
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-\draw [loosely dashed, thin] (axis cs: -0.05,2.0625) -- (axis cs: 2.5,2.0625);
-
-\draw [ultra thin] (axis cs: 2.8,-0.05) -- (axis cs: 2.8,2.46);
-\draw [ultra thin] (axis cs: -0.05,2.46) -- (axis cs: 2.8,2.46);
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-
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-\addplot[color=blue!80!black, only marks, style={mark=*}] coordinates {(2,1.5)};
-
-\addplot[color=blue, only marks, fill=white] coordinates {(2.5,2.0625)};
-
-\addplot[color=blue!80!black, only marks, style={mark=*}] coordinates {(2.8,2.46)};
-
-
-\node () at (axis cs:2.5,2.8) {$\displaystyle y=\left(x\right)$};
-
-\end{axis}
-
-
-\end{tikzpicture}

Band2/B2.5.tikz

@@ -1,58 +0,0 @@
-\begin{tikzpicture}
-[line cap=round,line join=round,x=1cm,y=1cm]
-\tikzset{
-        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
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-\begin{axis}[
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-axis lines=middle,
-axis line style = {-latex},
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-xmax=3,
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-ymax=2,
-ytick={1,2},
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-xlabel=$x$,
-ylabel=$y$,
-extra x ticks={0.5},
-    extra x tick style={
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-    extra x tick labels={$\frac{1}{2}$},
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-
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-\draw [thick, dashed] (axis cs: -0.05,1) -- (axis cs: 0.5,1);
-\draw [thick, dashed] (axis cs: 0.5,-0.05) -- (axis cs: 0.5,1);
-
-
-\addplot[domain=-1:0.5, blue, line width=1,samples=2000] {((x^2)-(1/4))/(x-(1/2)};
-\addplot[color=blue!80!black, only marks, style={mark=*, fill=white}] coordinates {(0.5,1)};
-\addplot[domain=0.51:1.5, blue, line width=1,samples=2000] {((x^2)-(1/4))/(x-(1/2)};
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-\draw [thick, dashed] (axis cs: 0.5,1) -- (axis cs: 0.5,2);
-
-
-
-\scriptsize{ \node () at (axis cs:2.4,1.8) {$\displaystyle y=\left\{ \begin{array}{rl}
-\frac{x^2-\frac{1}{4}}{x-\frac{1}{2}} & \text{für}\; x \neq \frac{1}{2}\\ \\
-2 & \text{für}\; x = \frac{1}{2}\\
-\end{array}
-\right .$};}
-
-% \draw[] (axis cs:1.3, 0.1) -- (axis cs:1.3, -0.1);
-% \node () at (axis cs:1.3,-0.25) {$x_0$};
-
-% \draw[] (axis cs:1.14,-0.05) -- (axis cs:1.14, 0.05);
-% \node () at (axis cs:0.33,-0.45) {$\frac{1}{\pi}$};
-
-\end{axis}
-
-
-\end{tikzpicture}

Band2/B2.6.tikz

@@ -1,33 +0,0 @@
-\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm]
-\begin{axis}[
-x=5cm,y=1.5cm,
-axis lines=middle,
-xmin=-1.2,
-xmax=1.2,
-ymin=-1.4,
-ymax=1.4,
-ytick={-1,...,1},
-xtick=\empty,
-xlabel=$x$,
-ylabel=$y$,
-]
-
-
-\addplot[domain=0.05:3.8, red, line width=1,samples=5000] {sin(deg(1/(x)))};
-\addplot[domain=-3.8:-0.05, red,  line width=1,samples=5000] {sin(deg(1/(x)))};
-
-%\draw (-0.32,0.1) node[anchor=north west] {$-\frac{1}{\pi}$};
-%\draw (0.32,0.1) node[anchor=north west] {$\frac{1}{\pi}$};
- \draw[] (axis cs:-0.32, 0.1) -- (axis cs:-0.32, -0.1);
- \node () at (axis cs:-0.35,-0.45) {$-\frac{1}{\pi}$};
-
-  \draw[] (axis cs:0.32, 0.1) -- (axis cs:0.32, -0.1);
- \node () at (axis cs:0.33,-0.45) {$\frac{1}{\pi}$};
-%\node[color=red, font=\footnotesize] at (-1,-0.25) {$f(x)=3x^3 - x^2 - 10x$};
-%\node[color=blue, font=\footnotesize] at (axis cs: 1.1,2.2) {$g(x)=- x^2 + 2x$};
-
-\end{axis}
-
-
-
-\end{tikzpicture}

Band2/B2.8.tikz

@@ -1,56 +0,0 @@
-\begin{tikzpicture}[line cap=round,line join=round,x=1cm,y=1cm]
-\tikzset{
-        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
-        small dot/.style={fill=black,circle,scale=0.3},}
-\begin{axis}[
-x=2cm,y=2cm,
-axis lines=middle,
-axis line style = {-latex},
-xmin=-0.5,
-xmax=3,
-ymin=-0.5,
-ymax=2,
-ytick=\empty,
-xtick=\empty,
-xlabel=$x$,
-ylabel=$y$,
-extra x ticks={1.3},
-    extra x tick style={
-        tick label style={anchor=north}},
-    extra x tick labels={$x_0$},
- extra y ticks={1.14},
- extra y tick style={
-        tick label style={anchor=east}},
-    extra y tick labels={$\displaystyle \sqrt{x_0}$},
-enlargelimits = true,
-]
-  \draw [thick, dashed] 
-        (axis cs: -0.05,1.14) -- (axis cs: 1.3,1.14);
-  \draw [thick, dashed] 
-        (axis cs: 1.3,-0.05) -- (axis cs: 1.3,1.14);
-\node[label={180:{}},circle,fill,inner sep=1.5] at (axis cs:1.3,1.14) {};
-\node[label={300:{$0$}},circle,fill,inner sep=1.5] at (axis cs:0,0) {};
- %       node[pos=0.5, above] {$y=12$};
- %  \addplot coordinates { (0,1.14) (1.3,1.14) };
- %   \addplot coordinates { (1,4) (2,6) };
-  %  \draw (axis cs:2,3) -- node[left]{Text} (axis cs:2,6);
-
-\addplot[domain=0:2, blue, line width=1,samples=5000] {sqrt(x)};
-%\addplot[domain=-3.8:-0.05, red,  line width=1,samples=5000] {sin(deg(1/(x)))};
-
-
-\begin{scriptsize}
- \node () at (axis cs:2.2,1.7) {$\displaystyle y=\sqrt{x}\;(x \neq 0)$};
-\end{scriptsize}
-
-% \draw[] (axis cs:1.3, 0.1) -- (axis cs:1.3, -0.1);
-% \node () at (axis cs:1.3,-0.25) {$x_0$};
-
-% \draw[] (axis cs:1.14,-0.05) -- (axis cs:1.14, 0.05);
-% \node () at (axis cs:0.33,-0.45) {$\frac{1}{\pi}$};
-
-\end{axis}
-
-
-
-\end{tikzpicture}

Band2/B2.9.tex

@@ -1,79 +0,0 @@
-%!tikz editor 1.0
-\documentclass{article}
-\usepackage{tikz}
-\usepackage[graphics, active, tightpage]{preview}
-\usepackage{circuitikz}
-\PreviewEnvironment{tikzpicture}
-
-%!tikz preamble begin
-\usepackage{pgfplots}
-%!tikz preamble end
-
-
-%%%%%%%%%
-%%   convert -density 300 GW001.pdf -quality 100 GW001.png
-%%%%%%%%%
-
-\begin{document}
-%!tikz source begin
-\begin{tikzpicture}
-[line cap=round,line join=round,x=1cm,y=1cm,scale=1]
-\tikzset{
-        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
-        small dot/.style={fill=black,circle,scale=0.3},}
-\begin{axis}[
-x=2cm,y=2cm,
-axis lines=middle,
-axis line style = {-latex},
-xmin=-0.5,
-xmax=3,
-ymin=-0.1,
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-ylabel=$y$,
-extra x ticks  = {0.75, 1.25,1.75,2.25},
-extra x tick labels= {$x_0$,  $x_3$, $x_2$, $x_1$},
-extra y ticks  = {0.71232, 1.22314, 1.55962, 1.81093},
-extra y tick labels= {$g_r$,  $f(x_3)$, $f(x_2)$, $f(x_1)$}
-]
-
-%\addplot[domain=-2.7:-1.1, blue, line width=1,samples=500] {1/(x^2-1)}; 
-%\addplot[domain=1.1:2.7, blue, line width=1,samples=500] {1/(x^2-1)}; 
-\addplot[domain=0.75:2.65, blue, line width=1,samples=500] {1+ln(x)}; 
-
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-\addplot[color=blue, only marks, fill=white] coordinates {(0.75,0.71232)};
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-
-\end{axis}
-
-
-\end{tikzpicture}
-%!tikz source end
-
-\end{document}

Band2/B2.9.tikz

@@ -1,56 +0,0 @@
-\begin{tikzpicture}
-[line cap=round,line join=round,x=1cm,y=1cm,scale=1]
-\tikzset{
-        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
-        small dot/.style={fill=black,circle,scale=0.3},}
-\begin{axis}[
-x=2cm,y=2cm,
-axis lines=middle,
-axis line style = {-latex},
-xmin=-0.5,
-xmax=3,
-ymin=-0.1,
-ymax=2.5,
-ytick=\empty,
-xtick=\empty,
-xlabel=$x$,
-ylabel=$y$,
-extra x ticks  = {0.75, 1.25,1.75,2.25},
-extra x tick labels= {$x_0$,  $x_3$, $x_2$, $x_1$},
-extra y ticks  = {0.71232, 1.22314, 1.55962, 1.81093},
-extra y tick labels= {$g_r$,  $f(x_3)$, $f(x_2)$, $f(x_1)$}
-]
-
-%\addplot[domain=-2.7:-1.1, blue, line width=1,samples=500] {1/(x^2-1)}; 
-%\addplot[domain=1.1:2.7, blue, line width=1,samples=500] {1/(x^2-1)}; 
-\addplot[domain=0.75:2.65, blue, line width=1,samples=500] {1+ln(x)}; 
-
-
-\addplot[color=blue, only marks, fill=white] coordinates {(0.75,0.71232)};
-
-\draw [dashed, blue] (axis cs: 0.75,-0.05) -- (axis cs: 0.75,0.71232);
-
-\draw [dashed, blue] (axis cs: -0.05,0.71232) -- (axis cs: 0.75,0.71232);
-
-\addplot[color=blue, only marks] coordinates {(1.25,1.22314)};
-
-
-\draw [ blue, thin] (axis cs: 1.25,-0.05) -- (axis cs: 1.25,1.22314);
-
-\draw [ blue, thin] (axis cs: -0.05, 1.22314) -- (axis cs: 1.25,1.22314);
-
-\addplot[color=blue, only marks] coordinates {(1.75,1.55962)};
-
-\draw [ blue, thin] (axis cs: 1.75,-0.05) -- (axis cs: 1.75,1.55962);
-
-\draw [ blue, thin] (axis cs: -0.05, 1.55962) -- (axis cs: 1.75,1.55962);
-
-\addplot[color=blue, only marks] coordinates {(2.25,1.81093)};
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-
-\draw [ blue, thin] (axis cs: 2.25,-0.05) -- (axis cs: 2.25,1.81093);
-
-\draw [ blue, thin] (axis cs: -0.05, 1.81093) -- (axis cs: 2.25,1.81093);
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-\end{axis}
-\end{tikzpicture}

Band2/Definitions.tex

@@ -205,6 +205,7 @@
 \newcounter{mybspcount}
 \newcounter{myaufgcount}
 \newcounter{mysatzcount}
+\newcounter{mybemcount}
 
 \newenvironment{definition}{%      define a custom environment
 %	\bigskip\noindent%         create a vertical offset to previous material
@@ -228,6 +229,12 @@
 \numberwithin{mybspcount}{chapter}
 
 
+\newenvironment{bemerkung}{%      define a custom environment
+	\bigskip\noindent%         create a vertical offset to previous material
+	\refstepcounter{mybemcount}% increment the environment's counter
+	\textbf{Bemerkung \themybemcount}}{}
+\numberwithin{mybemcount}{chapter}
+
 \newenvironment{aufgabe}{%      define a custom environment
 	\bigskip\noindent%         create a vertical offset to previous material
 	\refstepcounter{myaufgcount}% increment the environment's counter

Band2/Grenzwert.tex

@@ -454,17 +454,17 @@ d.h., nach hinreichend langer Zeit $t$ hat die Geschwindigkeit $v$ nahezu den ko
 
 Besitzt eine Funktion $f$ für eine der "`Bewegungen"'
 
-\begin{align}
+\begin{align}\label{eq:2.17}
 x \rightarrow x_0 ; \quad x \rightarrow x_0+0, x \rightarrow x_0-0 ; \quad x \rightarrow+\infty, x \rightarrow-\infty 
 \end{align}
 
-einen Grenzwert, dann heißt sie für diese "`Bewegung"' konvergent, andernfalls divergent. Wie für Zahlenfolgen kann man auch für Funktionen zwei Arten der Divergenz unterscheiden.
+einen Grenzwert, dann heißt sie für diese "`Bewegung"' \textit{konvergent}, andernfalls \textit{divergent}. Wie für Zahlenfolgen kann man auch für Funktionen zwei Arten der Divergenz unterscheiden.
 
 \begin{definition}\label{def:2.4}
-Die Funktion $f$ heißt\textbf{ bestimmt}\marginpar[\textbf{D.2.4}]{\textbf{D.2.4}} \textbf{divergent gegen} $+\infty(\text{bzw.}-\infty)$ für eine der "`Bewegungen"' (2.17) der unabhängigen Variablen $x$, wenn für jede diese "`Bewegung"' realisierende Folge\footnote{Man sagt z. B., die Folge $\left(x_n\right)$ \textit{realisiere} die "`Bewegung"' $x \rightarrow x_0+0$, wenn $x_n>x_0$ für alle $n$ und $\lim _{n \rightarrow \infty} x_n=x_0$ gilt.} $\left(x_n\right)$ in $D(f)$ die Folge $\left(f\left(x_n\right)\right)$ bestimmt divergent gegen $+\infty($ bzw. $-\infty)$ ist.
+Die Funktion $f$ heißt\textbf{ bestimmt}\marginpar[\textbf{D.2.4}]{\textbf{D.2.4}} \textbf{divergent gegen} $+\infty(\text{bzw.}-\infty)$ für eine der "`Bewegungen"' (\ref{eq:2.17}) der unabhängigen Variablen $x$, wenn für jede diese "`Bewegung"' realisierende Folge\footnote{Man sagt z. B., die Folge $\left(x_n\right)$ \textit{realisiere} die "`Bewegung"' $x \rightarrow x_0+0$, wenn $x_n>x_0$ für alle $n$ und $\lim _{n \rightarrow \infty} x_n=x_0$ gilt.} $\left(x_n\right)$ in $D(f)$ die Folge $\left(f\left(x_n\right)\right)$ bestimmt divergent gegen $+\infty($ bzw. $-\infty)$ ist.
 
 
-Ist $f$ für eine der "`Bewegungen"' (2.17) weder konvergent noch bestimmt divergent, so heißt $f$ für diese "`Bewegung"' \textbf{unbestimmt divergent}.
+Ist $f$ für eine der "`Bewegungen"' (\ref{eq:2.17}) weder konvergent noch bestimmt divergent, so heißt $f$ für diese "`Bewegung"' \textbf{unbestimmt divergent}.
 \end{definition}
 
 
@@ -476,56 +476,58 @@ und sagt auch, $f$ habe für $x \rightarrow x_0$ den \textit{uneigentlichen Gren
 
 \begin{beispiel} \label{bsp:2.12}
 
+Es gilt
+$$
+\lim _{x \rightarrow 0} \frac{1}{x^2}=+\infty
+$$
+(s. Bild \ref{fig:b2.2.12}), denn in Band 1, Beispiel 10.11, wurde gezeigt, daß für jede Folge $\left(x_n\right)$ mit $x_n \neq 0$ für alle $n$ und $\lim _{n \rightarrow \infty} x_n=0$ die Folge $\left(\frac{1}{x_n^2}\right)$ bestimmt divergent gegen $+\infty$ ist.
 
+\end{beispiel}
+
+\begin{beispiel} \label{bsp:2.13}
+
+Es soll die Grenzwertaussage
+$$
+\lim _{x \rightarrow+0} \ln x=-\infty
+$$
+bewiesen werden. Es sei $\left(x_n\right)$ eine Nullfolge mit $x_n>0$ für alle $n$. Zu jeder (insbesondere beliebig großen) Zahl $K>0$ existiert dann eine natürliche Zahl $n_0=n_0(K)$, so daß gilt 
+$$
+\begin{array}{lll}
+	&x_n=\left|x_n-0\right|<\mathrm{e}^{-K} & \text { für alle } n \geqq n_0,  \\
+	\text{also } & &\\
+	&\ln x_n<-K                             & \text { für alle } n \geqq n_0 .
+\end{array}
+$$
+Daraus folgt $\lim _{n \rightarrow \infty} \ln x_n=-\infty$, und die Behauptung ist bewiesen.
+
+\end{beispiel}
+
+\begin{beispiel} \label{bsp:2.14}
+
+Die Funktion $f(x)=\sin x$ ist für $x \rightarrow+\infty$ unbestimmt divergent. Zum Beweis dieser Behauptung betrachten wir die Folge $\left(x_n\right)$ mit $x_n=n \pi-\frac{\pi}{2}(n=1$, $2, \ldots)$ : Offenbar gilt $\lim _{n \rightarrow \infty} x_n=+\infty$, aber wegen $f\left(x_n\right)=(-1)^{n+1}$ ist die Folge $\left(f\left(x_n\right)\right)$ unbestimmt divergent. Ganz entsprechend hatten wir bereits in Beispiel 2.4 gezeigt, daß die Funktion $f(x)=\sin \frac{1}{x}(x \neq 0$ ) für $x \rightarrow 0$ unbestimmt divergent ist.
 	
 \end{beispiel}
 
 
+\section{Grenzwertsätze}
 
-
-HIERHIERHIERHIERHIER
-
-
-Beispiel 2.12: Es gilt
-$$
-	\lim _{x \rightarrow 0} \frac{1}{x^2}=+\infty
-$$
-(s. Bild 2.12), denn in Band 1, Beispiel 10.11, wurde gezeigt, daß für jede Folge $\left(x_n\right)$ mit $x_n \neq 0$ für alle $n$ und $\lim _{n \rightarrow \infty} x_n=0$ die Folge $\left(\frac{1}{x_n^2}\right)$ bestimmt divergent gegen $+\infty$ ist.
-Beispiel 2.13: Es soll die Grenzwertaussage
-$$
-	\lim _{x \rightarrow+0} \ln x=-\infty
-$$
-bewiesen werden. Es sei $\left(x_n\right)$ eine Nullfolge mit $x_n>0$ für alle $n$. Zu jeder (insbeson
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-
-dere beliebig großen) Zahl $K>0$ existiert dann eine natürliche Zahl $n_0=n_0(K)$, so daß gilt also
-$$
-	\begin{array}{ll}
-		x_n=\left|x_n-0\right|<\mathrm{e}^{-K} & \text { für alle } n \geqq n_0,  \\
-		\ln x_n<-K                             & \text { für alle } n \geqq n_0 .
-	\end{array}
-$$
-Daraus folgt $\lim _{n \rightarrow \infty} \ln x_n=-\infty$, und die Behauptung ist bewiesen.
-Beispiel 2.14: Die Funktion $f(x)=\sin x$ ist für $x \rightarrow+\infty$ unbestimmt divergent. Zum Beweis dieser Behauptung betrachten wir die Folge $\left(x_n\right)$ mit $x_n=n \pi-\frac{\pi}{2}(n=1$, $2, \ldots)$ : Offenbar gilt $\lim _{n \rightarrow \infty} x_n=+\infty$, aber wegen $f\left(x_n\right)=(-1)^{n+1}$ ist die Folge $\left(f\left(x_n\right)\right)$ unbestimmt divergent. Ganz entsprechend hatten wir bereits in Beispiel 2.4 gezeigt, daß die Funktion $f(x)=\sin \frac{1}{x}(x \neq 0$ ) für $x \rightarrow 0$ unbestimmt divergent ist.
-
-	2.5. Grenzwertsätze
 In diesem Abschnitt werden einige Regeln für das Rechnen mit Grenzwerten von Funktionen angegeben. Da der Grenzwertbegriff für Funktionen auf den Grenzwertbegriff für Zahlenfolgen zurückgeführt wurde, kann man diese Regeln leicht aus den entsprechenden Grenzwertsätzen für Zahlenfolgen ableiten. Wir verzichten auf eine Durchführung der Beweise.
 	
-	Bemerkung 2.1: Die folgenden für die ,,Bewegung“ $x \rightarrow x_0$ formulierten Sätze gelten sinngemä $\beta^1$ ) auch für die ,,Bewegungen“
+	
+\begin{bemerkung}
+ Die folgenden für die "`Bewegung"' $x \rightarrow x_0$ formulierten Sätze gelten sinngemäß \footnote{1) Wird z. B. statt $x \rightarrow x_0$ die "`Bewegung"' $x \rightarrow+\infty$ betrachtet, so ist in den folgenden Sätzen ,"`punktierte Umgebung von $x_0$ "' durch "`Intervall $(a,+\infty)$"' zu ersetzen. Analog ist in den anderen Fällen zu verfahren.} auch für die "`Bewegungen"'
+ 
 $$
 x \rightarrow x_0+0, x \rightarrow x_0-0 ; \quad x \rightarrow+\infty, x \rightarrow-\infty .
 $$
-	Satz 2.3: Die Funktionen $f_1$ und $f_2$ seien für $x \rightarrow x_0$ konvergent mit
-	S. 2.3
+\end{bemerkung}
+
+\begin{satz}
+Die Funktionen $f_1$ und $f_2$ seien\reversemarginpar\marginpar[\textbf{S.2.3}]{\textbf{S.2.3}} \label{satz:2.3}  für $x \rightarrow x_0$ konvergent mit 
 $$
 \lim _{x \rightarrow x_0} f_1(x)=g_1, \quad \lim _{x \rightarrow x_0} f_2(x)=g_2 .
 $$	
 
-
 Dann gilt
 $$
 \begin{aligned}
@@ -539,7 +541,33 @@ Beispiel 2.14: Die Funktion $f(x)=\sin x$ ist für $x \rightarrow+\infty$ unbest
 $$
 \lim _{x \rightarrow x_0} \frac{f_1(x)}{f_2(x)}=\frac{g_1}{g_2} .
 $$
-	1) Wird z. B. statt $x \rightarrow x_0$ die „Bewegung“ $x \rightarrow+\infty$ betrachtet, so ist in den folgenden Sätzen ,,punktierte Umgebung von $x_0$ " durch „Intervall $(a,+\infty)$ " zu ersetzen. Analog ist in den anderen Fällen zu verfahren.
+\end{satz}
+
+\begin{beispiel} \label{bsp:2.15}
+	
+	
+\end{beispiel}
+
+
+
+HIERHIERHIERHIERHIER
+
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+	
+
+
+
+
+
+	
 
 
 	Beispiel 2.15: Gesucht ist der Grenzwert

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-\draw ($0.5*(m1-2-1.south west)+0.5*(m1-3-1.north west)$) -- ([xshift=5mm]$0.5*(m1-2-7.south east)+0.5*(m1-3-7.north east)$);
-
-\draw ([yshift=1mm]m1-3-7.north west) -- (m1-3-7.south west);
-\draw (m1-3-7.south west) -- (m1-3-7.south east);
-
-\draw ($0.5*(m2-1-1.south west)+0.5*(m2-2-1.north west)$) -- ([xshift=5mm]$0.5*(m2-1-6.south east)+0.5*(m2-2-6.north east)$);
-
-\draw ([yshift=1mm]m2-2-6.north west) -- (m2-2-6.south west);
-\draw (m2-2-6.south west) -- (m2-2-6.south east);
-
-\draw ([yshift=1.5mm]$0.5*(m3-1-1.south west)+0.5*(m3-2-1.north west)$) -- ([xshift=5mm]$0.5*(m3-1-5.south east)+0.5*(m3-2-5.north east)$);
-
-\draw ([yshift=1mm]m3-2-5.north west) -- (m3-2-5.south west);
-\draw (m3-2-5.south west) -- (m3-2-5.south east);
-
-\draw ([yshift=1.5mm]$0.5*(m4-1-1.south west)+0.5*(m4-2-1.north west)$) -- ([xshift=12mm]$0.5*(m4-1-4.south east)+0.5*(m4-2-4.north east)$);
-
-\draw ([yshift=1mm]m4-2-4.north west) -- (m4-2-4.south west);
-\draw (m4-2-4.south west) -- (m4-2-4.south east);
-
-\draw([xshift=2mm]m1-1-1.north east)--([xshift=2mm,yshift=-2mm]m4-2-1.south east);
-
-\end{tikzpicture}