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-\draw [thick, dashed] (axis cs: 0.5,-0.05) -- (axis cs: 0.5,1); - - -\addplot[domain=-2:2, blue, line width=1,samples=2000] {x^2}; - -\addplot[color=blue!80!black, only marks, style={mark=*}] coordinates {(0.5,2)}; - -\draw [ultra thin, dashed] (axis cs: -.05,1/4) -- (axis cs: 1/2,1/4); -\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} -%!tikz source end - -\end{document} \ No newline at end of file diff --git a/Band2/B2.10.tikz b/Band2/B2.10.tikz deleted file mode 100644 index 64f3c2b..0000000 --- a/Band2/B2.10.tikz +++ /dev/null @@ -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, - every axis x label/.style={at={(current axis.right of origin)},anchor=north east}, - 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$y=\left\{\begin{array}{l} -\frac{3}{x} \text { für } 03 -\end{array}\right.$}; - -\end{axis} -\end{tikzpicture} \ No newline at end of file diff --git a/Band2/B2.11.tikz b/Band2/B2.11.tikz deleted file mode 100644 index 7b1d493..0000000 --- a/Band2/B2.11.tikz +++ /dev/null @@ -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} \ No newline at end of file diff --git a/Band2/B2.12.tikz b/Band2/B2.12.tikz deleted file mode 100644 index 5142d24..0000000 --- a/Band2/B2.12.tikz +++ /dev/null @@ -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} \ No newline at end of file diff --git a/Band2/B2.19.png b/Band2/B2.19.png deleted file mode 100644 index 225e381..0000000 Binary files a/Band2/B2.19.png and /dev/null differ diff --git a/Band2/B2.3.tikz b/Band2/B2.3.tikz deleted file mode 100644 index c2038b7..0000000 --- a/Band2/B2.3.tikz +++ /dev/null @@ -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=$x$, -extra x ticks={1, 2, 3}, -extra x tick labels={$x_0 - C$, $x_0$, $x_0+C$}, -] -\draw [line width=2] (axis cs: 1,0) -- (axis cs: 3,0); -\addplot[only marks, fill=white, thick] coordinates {(1,0)}; -\addplot[only marks, fill=white, thick] coordinates {(2,0)}; -\addplot[only marks, fill=white, thick] coordinates {(3,0)}; -\end{axis} - -\end{tikzpicture} \ No newline at end of file diff --git a/Band2/B2.4.tikz b/Band2/B2.4.tikz deleted file mode 100644 index 26a04b1..0000000 --- a/Band2/B2.4.tikz +++ /dev/null @@ -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},} -\begin{axis}[ -x=2cm,y=2cm, -axis lines=middle, -axis line style = {-latex}, -xmin=-0.75, -xmax=3.75, -ymin=-0.5, -ymax=3.5, -ytick=\empty, -xtick=\empty, -xlabel=$x$, -ylabel=$y$, -extra x ticks={0.75, 2, 2.5,2.8}, -extra x tick labels={$x_1$, $x_3$, $x_0$, $x_2$}, -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); - -\draw [ultra thin] (axis cs: 2,-0.05) -- (axis cs: 2,1.5); -\draw [ultra thin] (axis cs: -0.05,1.5) -- (axis cs: 2,1.5); - -\draw [loosely dashed, ultra thin] (axis cs: 2.5,-0.05) -- (axis cs: 2.5,2.0625); -\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); - - -\addplot[color=blue!80!black, only marks, style={mark=*}] coordinates {(0.75,0.640625)}; -\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} diff --git a/Band2/B2.5.tikz b/Band2/B2.5.tikz deleted file mode 100644 index 3b6e682..0000000 --- a/Band2/B2.5.tikz +++ /dev/null @@ -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}, - small dot/.style={fill=black,circle,scale=0.3},} -\begin{axis}[ -x=2cm,y=2cm, -axis lines=middle, -axis line style = {-latex}, -xmin=-1, -xmax=3, -ymin=-0.5, -ymax=2, -ytick={1,2}, -yticklabels={1,2}, -xtick=\empty, -xlabel=$x$, -ylabel=$y$, -extra x ticks={0.5}, - extra x tick style={ - tick label style={anchor=north}}, - extra x tick labels={$\frac{1}{2}$}, -enlargelimits = true, -] - - - -\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)}; - -\addplot[color=blue!80!black, only marks, style={mark=*}] coordinates {(0.5,2)}; - -\draw [thick, dashed] (axis cs: -0.05,2) -- (axis cs: 0.5,2); -\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} \ No newline at end of file diff --git a/Band2/B2.6.tikz b/Band2/B2.6.tikz deleted file mode 100644 index 4679aa0..0000000 --- a/Band2/B2.6.tikz +++ /dev/null @@ -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} \ No newline at end of file diff --git a/Band2/B2.8.tikz b/Band2/B2.8.tikz deleted file mode 100644 index e0285c7..0000000 --- a/Band2/B2.8.tikz +++ /dev/null @@ -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} \ No newline at end of file diff --git a/Band2/B2.9.tex b/Band2/B2.9.tex deleted file mode 100644 index 0df0b6c..0000000 --- a/Band2/B2.9.tex +++ /dev/null @@ -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, -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) -- (axis cs: 1.25,1.22314); - -\draw [ blue, thin] (axis cs: 0, 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) -- (axis cs: 1.75,1.55962); - -\draw [ blue, thin] (axis cs: 0, 1.55962) -- (axis cs: 1.75,1.55962); - -\addplot[color=blue, only marks] coordinates {(2.25,1.81093)}; - - -\draw [ blue, thin] (axis cs: 2.25,0) -- (axis cs: 2.25,1.81093); - -\draw [ blue, thin] (axis cs: 0, 1.81093) -- (axis cs: 2.25,1.81093); - -\end{axis} - - -\end{tikzpicture} -%!tikz source end - -\end{document} \ No newline at end of file diff --git a/Band2/B2.9.tikz b/Band2/B2.9.tikz deleted file mode 100644 index 9a34262..0000000 --- a/Band2/B2.9.tikz +++ /dev/null @@ -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)}; - - -\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); - -\end{axis} -\end{tikzpicture} diff --git a/Band2/Definitions.tex b/Band2/Definitions.tex index b632ac0..ef300a6 100644 --- a/Band2/Definitions.tex +++ b/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 diff --git a/Band2/Grenzwert.tex b/Band2/Grenzwert.tex index fcb3182..eb0cc50 100644 --- a/Band2/Grenzwert.tex +++ b/Band2/Grenzwert.tex @@ -196,7 +196,7 @@ Man kann den Beweis auch dadurch führen, daß man zwei Folgen $\left(x_n\right) \lim _{n \rightarrow \infty} f\left(x_n\right)=\mathrm{e} . \end{align} - Ohne Beweis \footnote{Beweise zu Teil 1 dieses Buches findet man, wenn nichts anderes gesagt ist, in [5] und [10].} sei mitgeteilt, daß (2.6) sogar für jede Folge $\left(x_n\right)$ mit $x_n>-1, x_n \neq 0$ und $\lim _{n \rightarrow \infty} x_n=0$ gilt. Damit erhält man den für spätere Anwendungen wichtigen Grenzwert + Ohne Beweis\footnote{Beweise zu Teil 1 dieses Buches findet man, wenn nichts anderes gesagt ist, in [5] und [10].} sei mitgeteilt, daß (2.6) sogar für jede Folge $\left(x_n\right)$ mit $x_n>-1, x_n \neq 0$ und $\lim _{n \rightarrow \infty} x_n=0$ gilt. Damit erhält man den für spätere Anwendungen wichtigen Grenzwert \begin{align} \lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=\mathrm{e} \end{align} @@ -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} -x \rightarrow x_0 ; \quad x \rightarrow x_0+0, x \rightarrow x_0-0 ; \quad x \rightarrow+\infty, x \rightarrow-\infty +\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} @@ -476,70 +476,98 @@ 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} + +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. + + +\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"' + +$$ +x \rightarrow x_0+0, x \rightarrow x_0-0 ; \quad x \rightarrow+\infty, x \rightarrow-\infty . +$$ +\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 +$$ +\begin{aligned} + & \lim _{x \rightarrow x_0}\left[f_1(x)+f_2(x)\right]=g_1+g_2, \\ + & \lim _{x \rightarrow x_0}\left[f_1(x)-f_2(x)\right]=g_1-g_2, \\ + & \lim _{x \rightarrow x_0}\left[c f_1(x)\right]=c g_1 \quad(c \text { eine Konstante }), \\ + & \lim _{x \rightarrow x_0}\left[f_1(x) \cdot f_2(x)\right]=g_1 \cdot g_2 . +\end{aligned} +$$ +Ist außerdem $f_2(x) \neq 0$ für alle $x$ einer punktierten Umgebung von $x_0$ und $g_2 \neq 0$, dann gilt auch +$$ +\lim _{x \rightarrow x_0} \frac{f_1(x)}{f_2(x)}=\frac{g_1}{g_2} . +$$ +\end{satz} + +\begin{beispiel} \label{bsp:2.15} + + +\end{beispiel} 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“ - $$ - x \rightarrow x_0+0, x \rightarrow x_0-0 ; \quad x \rightarrow+\infty, x \rightarrow-\infty . - $$ - Satz 2.3: Die Funktionen $f_1$ und $f_2$ seien für $x \rightarrow x_0$ konvergent mit - S. 2.3 - $$ - \lim _{x \rightarrow x_0} f_1(x)=g_1, \quad \lim _{x \rightarrow x_0} f_2(x)=g_2 . - $$ + - Dann gilt - $$ - \begin{aligned} - & \lim _{x \rightarrow x_0}\left[f_1(x)+f_2(x)\right]=g_1+g_2, \\ - & \lim _{x \rightarrow x_0}\left[f_1(x)-f_2(x)\right]=g_1-g_2, \\ - & \lim _{x \rightarrow x_0}\left[c f_1(x)\right]=c g_1 \quad(c \text { eine Konstante }), \\ - & \lim _{x \rightarrow x_0}\left[f_1(x) \cdot f_2(x)\right]=g_1 \cdot g_2 . - \end{aligned} - $$ - Ist außerdem $f_2(x) \neq 0$ für alle $x$ einer punktierten Umgebung von $x_0$ und $g_2 \neq 0$, dann gilt auch - $$ - \lim _{x \rightarrow x_0} \frac{f_1(x)}{f_2(x)}=\frac{g_1}{g_2} . - $$ - 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. 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Schritt}; - - %\draw ($(m1-2-5.north east)!0.5!(m1-2-5.north west)$) -- ($(m1-2-5.south east)!0.5!(m2-2-2.south west)$); - %\draw($(m1-2-5.north east)!50!(m1-2-6.north west)$)--($(m1-3-5.south east)!50!(m1-3-6.south west)$); - %\draw([yshift=3mm]m1-3-6.north west)--(m1-3-6.south west); -%\draw ([yshift=-3mm]m1-2-1.south west) -- ([yshift=-3mm,xshift=10mm]m1-2-6.south east); - -\draw ($0.5*(m1-2-1.south west)+0.5*(m1-3-1.north west)$) -- ([xshift=5mm]$0.5*(m1-2-6.south east)+0.5*(m1-3-6.north east)$); - -\draw ([yshift=1mm]m1-3-6.north west) -- (m1-3-6.south west); -\draw (m1-3-6.south west) -- (m1-3-6.south east); - -\draw ($0.5*(m2-1-1.south west)+0.5*(m2-2-1.north west)$) -- ([xshift=5mm]$0.5*(m2-1-5.south east)+0.5*(m2-2-5.north east)$); - -%Ende Zeile 2 -%vert -\draw ([yshift=1mm]m2-2-5.north west) -- (m2-2-5.south west); -%horiz -\draw (m2-2-5.south west) -- (m2-2-5.south east); - - -%Trennlinie -\draw ([yshift=1.5mm]$0.5*(m3-1-1.south west)+0.5*(m3-2-1.north west)$) -- ([xshift=5mm]$0.5*(m3-1-4.south east)+0.5*(m3-2-4.north east)$); -\node[circle] (del-left-2) at ($0.5*(m2-2-1.south west)+0.5*(m2-1-1.north west)$){}; -\node[left=10pt] at (del-left-2.west) {2. Schritt}; - -\node[circle] (del-left-3) at ([yshift=-1mm]$0.5*(m3-1-1.south west)+0.5*(m3-2-1.north west)$){}; -\node[left=10pt] at (del-left-3.west) {3. Schritt}; - -\draw ([yshift=1mm]m3-2-4.north west) -- (m3-2-4.south west); -\draw (m3-2-4.south west) -- (m3-2-4.south east); - -%\draw ([yshift=-3mm,thick]m2-1-1.south west) -- ([yshift=-3mm,xshift=10mm]m2-1-5.south east); - -\draw ([yshift=1.5mm]$0.5*(m4-1-1.south west)+0.5*(m4-2-1.north west)$) -- ([xshift=5mm]$0.5*(m4-1-3.south east)+0.5*(m4-2-3.north east)$); - -\draw ([yshift=1mm]m4-2-3.north west) -- (m4-2-3.south west); -\draw (m4-2-3.south west) -- (m4-2-3.south east); - - -\node[circle] (del-left-4) at ([yshift=-1mm]$0.5*(m4-1-1.south west)+0.5*(m4-2-1.north west)$){}; -\node[left=10pt] at (del-left-4.west) {4. Schritt}; - -\draw ($0.5*(m1-2-1.south west)+0.5*(m1-3-1.north west)$) -- ([xshift=5mm]$0.5*(m1-2-6.south east)+0.5*(m1-3-6.north east)$); - -\draw ([yshift=1mm]m1-3-6.north west) -- (m1-3-6.south west); -\draw (m1-3-6.south west) -- (m1-3-6.south east); - -\draw ($0.5*(m2-1-1.south west)+0.5*(m2-2-1.north west)$) -- ([xshift=5mm]$0.5*(m2-1-5.south east)+0.5*(m2-2-5.north east)$); - - -\draw ([yshift=1mm]m3-2-4.north west) -- (m3-2-4.south west); -\draw (m3-2-4.south west) -- (m3-2-4.south east); - -%\draw ([yshift=-3mm,thick]m2-1-1.south west) -- ([yshift=-3mm,xshift=10mm]m2-1-5.south east); - -\node[circle] (del-left-5) at ([yshift=1mm]$0.5*(m5-1-1.south west)+0.5*(m5-1-1.north west)$){}; -\node[left=10pt] at (del-left-5.west) {5. Schritt}; - - -\draw (m5-1-2.north west) -- (m5-1-2.south west); -\draw (m5-1-2.south west) -- (m5-1-2.south east); - - -\draw ([yshift=5.5mm]m5-1-1.north west)-- ([yshift=1mm]m5-1-2.north east); - -\draw(m1-1-1.north east)--([yshift=-2mm]m5-1-1.south east); - - %\mymatrixbracetop{2}{6}{$E'$} - - %\node[font=\color{red}] at (m2.center){X}; - - %\foreach \xy in {$1*(m2-1-1)$, $1*(m2-2-1)$}{ - % \node at (\xy) {\xy}; - %} - -\end{tikzpicture} \footnotetext{Man beachte, ...} - -\newpage diff --git a/Band2/Horner02.tikz b/Band2/Horner02.tikz deleted file mode 100644 index 844fe43..0000000 --- a/Band2/Horner02.tikz +++ /dev/null @@ -1,56 +0,0 @@ -%!TEX root=Band2.tex -\begin{tikzpicture}[>=latex] - -%\draw[cyan, densely dotted] (-2,0) grid (12,14); -\useasboundingbox (-2,0) rectangle (12,14); - -%nodes=draw, -\matrix (m1) [anchor=west,row sep=2mm,column sep=5mm,matrix of math nodes,column 7/.style={anchor=base west}] at (1, 12) - { - \phantom{a} & 1 & 3 & 5 & 7 & 6 & 2 \\ - \phantom{a} & \phantom{a} & -1 & -2 & -3 & -4 & -2\\ - -1 & 1 & 2 & 3 & 4 & 2 & 0=g(-1)\\ - }; - -\matrix (m2) [anchor=west,row sep=2mm,column sep=5mm,matrix of math nodes, column 6/.style={anchor=base west}] at (1, 10) - { - \phantom{a} & \phantom{a} & -1 & -1 & -2 & -2\\ - -1 & 1 & 1 & 2 & 2 & 0=g'(-1) \\ - }; - -\matrix (m3) [anchor=west,row sep=2mm,column sep=5mm,matrix of math nodes,column 5/.style={anchor=base west}] at (1, 8.1) -{ - \phantom{a} & \phantom{a} & -1 & 0 & -2\\ - -1 & 1 & 0 & 2 & 0=\frac{g''(-1)}{2!} \\ -}; - -\matrix (m4) [anchor=west,row sep=2mm,column sep=5mm,matrix of math nodes,column 4/.style={anchor=base west}] at (1, 6) -{ - \phantom{a} & \phantom{a} & -1 & 1\\ - -1 & 1 & -1 & 3=\frac{g'''(-1)}{3!} \neq 0 \\ -}; - - -\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}