2019-2020/Tsti2d/Analyse/Limites/3B_fonction_reference.tex

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\documentclass[a4paper,10pt]{article}
\usepackage{myXsim}
\title{Fonctions de référence - Limites}
\tribe{Terminale Sti2d}
\date{Novembre 2019}
%\geometry{left=10mm,right=10mm, top=10mm}
%\pagestyle{empty}
\begin{document}
\setcounter{section}{2}
\section{Fonctions de référence}
\begin{itemize}
\item Fonction carré $x\mapsto x^2$
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[yscale=.5, xscale=.8]
\tkzInit[xmin=-4,xmax=4,xstep=1,
ymin=0,ymax=10,ystep=1]
\tkzGrid
\tkzAxeXY[up space=0.5,right space=.5]
\tkzFct[domain = -5:5, line width=1pt]{x**2}
\tkzText[draw,fill = brown!20](2.5,1){$f(x)=x^2$}
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}
\tkzTabInit[lgt=2,espcl=3]{$x$/1,$f(x)$/3}%
{$-\infty$, $0$, $+\infty$}%
\tkzTabVar{+/$+\infty$, -/0, +/$+\infty$}%
\end{tikzpicture}
Limites
\[
\lim_{x\rightarrow-\infty} x^2 = +\infty \qquad
\lim_{x\rightarrow+\infty} x^2 = +\infty
\]
\end{minipage}
\item Fonction cube $x\mapsto x^3$
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[yscale=0.5, xscale=1]
\tkzInit[xmin=-3,xmax=3,xstep=1,
ymin=-10,ymax=10,ystep=2]
\tkzGrid
\tkzAxeXY[up space=0.5,right space=.5]
\tkzFct[domain = -3:3, line width=1pt]{x**3}
\tkzText[draw,fill = brown!20](2,-8){$f(x)=x^3$}
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}
\tkzTabInit[lgt=2,espcl=5]{$x$/1,$f(x)$/3}%
{$-\infty$, $+\infty$}%
\tkzTabVar{-/$-\infty$, +/$+\infty$}%
\tkzTabVal{1}{2}{0.5}{0}{0}
\end{tikzpicture}
Limites
\[
\lim_{x\rightarrow-\infty} x^3 = -\infty \qquad
\lim_{x\rightarrow+\infty} x^3 = +\infty
\]
\end{minipage}
\item Fonction inverse $x \mapsto \dfrac{1}{x}$
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[yscale=.5, xscale=.8]
\tkzInit[xmin=-4,xmax=4,xstep=1,
ymin=-5,ymax=5,ystep=1]
\tkzGrid
\tkzAxeXY[up space=0.5,right space=.5]
\tkzFct[domain = -5:-0.01, line width=1pt]{1/x}
\tkzFct[domain = 0.01:5, line width=1pt]{1/x}
\tkzText[draw,fill = brown!20](3,-4){$f(x)=\frac{1}{x}$}
\tkzHLine[color=red,style=solid,line width=1.2pt]{0}
\tkzVLine[color=green,style=solid,line width=1.2pt]{0}
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}
\tkzTabInit[lgt=1.5,espcl=3]{$x$ /1,$f(x)$ /3}
{$-\infty$,$0$,$+\infty$}%
\tkzTabVar{+/
$0$ / ,-D+/ $-\infty$ / $+\infty$ , -/ $0$ /}
\end{tikzpicture}
\end{minipage}
Limites
\[
\lim_{x\rightarrow-\infty} \frac{1}{x} = 0 \qquad
\lim_{x\rightarrow 0^-} \frac{1}{x} = -\infty \qquad
\lim_{x\rightarrow 0^+} \frac{1}{x} = +\infty \qquad
\lim_{x\rightarrow+\infty} \frac{1}{x} = 0
\]
\textbf{Asymptote horizontale} en $-\infty$ et $+\infty$ d'équation $y=0$ (en rouge)\\
\textbf{Asymptote verticale} en $0^-$ et $0^+$ d'équation $x=0$ (en vert).
\pagebreak
\item Fonction exponentielle $x\mapsto e^x$
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[yscale=1, xscale=.8]
\tkzInit[xmin=-5,xmax=2,xstep=1,
ymin=0,ymax=5,ystep=1]
\tkzGrid
\tkzAxeXY[up space=0.5,right space=.5]
\tkzFct[domain = -5:2, line width=1pt]{exp(x)}
\tkzText[draw,fill = brown!20](2,0.5){$f(x)=\text{e}^{x}$}
\tkzHLine[color=red,style=solid,line width=1.2pt]{0}
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}
\tkzTabInit[lgt=2,espcl=5]{$x$/1,$f(x)$/3}%
{$-\infty$, $+\infty$}%
\tkzTabVar{-/$0$, +/$+\infty$}%
\end{tikzpicture}
Limites
\[
\lim_{x\rightarrow-\infty} e^x = 0 \qquad
\lim_{x\rightarrow+\infty} e^x = +\infty
\]
\end{minipage}
\textbf{Asymptote horizontale} en $-\infty$ d'équation $y=0$ (en rouge)\\
\item Fonction logarithme népérien $x \mapsto \ln{x}$
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[yscale=0.8, xscale=1]
\tkzInit[xmin=0,xmax=6,xstep=1,
ymin=-3,ymax=3,ystep=1]
\tkzGrid
\tkzAxeXY[up space=0.5,right space=.5]
\tkzFct[domain = 0.01:6, line width=1pt]{log(x)}
\tkzText[draw,fill = brown!20](5,-2.5){$f(x)=\ln(x)$}
\tkzVLine[color=green,style=solid,line width=1.2pt]{0}
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}
\tkzTabInit[lgt=2,espcl=5]{$x$/1,$f(x)$/3}%
{$0$, $+\infty$}%
\tkzTabVar{D-/$-\infty$, +/$+\infty$}%
\end{tikzpicture}
Limites
\[
\lim_{x\rightarrow 0} \ln{x} = -\infty \qquad
\lim_{x\rightarrow+\infty} \ln{x} = +\infty
\]
\end{minipage}
\textbf{Asymptote verticale} en $0$ d'équation $x=0$ (en vert)\\
\end{itemize}
\end{document}