2020-2021/TST_sti2d/09_Limites_de_fonctions/1B_fonction_reference.tex

\documentclass[a4paper,10pt]{article}
\usepackage{myXsim}

\author{Benjamin Bertrand}
\title{Limites de fonctions - Cours}
\date{avril 2021}

\pagestyle{empty}

\begin{document}

\maketitle

\section{Tableaux de variations et limites des 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
        \[
            \lim_{x\rightarrow-\infty} \frac{1}{x} = 0 \qquad
            \lim_{\substack{x\rightarrow 0 \\ <}} \frac{1}{x} = -\infty \qquad
            \lim_{\substack{x\rightarrow 0 \\ >}} \frac{1}{x} = +\infty \qquad
            \lim_{x\rightarrow+\infty} \frac{1}{x} = 0
        \]

        \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}


    \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
            \[
                \lim_{\substack{x\rightarrow 0\\ > }} \ln{x} = -\infty \qquad
                \lim_{x\rightarrow+\infty} \ln{x} = +\infty
            \]
        \end{minipage}

\end{itemize}

\end{document}