2020-2021/TST_sti2d/09_Limites_de_fonctions/exercises.tex

\collectexercises{banque}
\begin{exercise}[subtitle={Limites de fonctions}, step={1}, origin={Création}, topics={Limites de fonctions}, tags={Fonctions, limites}]
\begin{tikzpicture}[yscale=.5, xscale=.8]
    \tkzInit[xmin=-5,xmax=5,xstep=1,
    ymin=0,ymax=10,ystep=1]
    \tkzGrid
    \tkzAxeXY
    \tkzFct[domain = -5:5, line width=1pt]{x**2}
    \tkzText[draw,fill = brown!20](3,1){$f(x)=x^2$}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[yscale=0.5, xscale=1]
    \tkzInit[xmin=-4,xmax=4,xstep=1,
    ymin=-10,ymax=10,ystep=2]
    \tkzGrid
    \tkzAxeXY
    \tkzFct[domain = -5:5, line width=1pt]{x**3}
    \tkzText[draw,fill = brown!20](1,-2){$f(x)=x^3$}
\end{tikzpicture}

\begin{tikzpicture}[yscale=1, xscale=.8]
    \tkzInit[xmin=-5,xmax=5,xstep=1,
    ymin=0,ymax=5,ystep=1]
    \tkzGrid
    \tkzAxeXY
    \tkzFct[domain = -5:5, line width=1pt]{exp(x)}
    \tkzText[draw,fill = brown!20](2,1){$f(x)=\text{e}^{x}$}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[yscale=1, xscale=1.5]
    \tkzInit[xmin=0,xmax=5,xstep=1,
    ymin=-3,ymax=3,ystep=1]
    \tkzGrid
    \tkzAxeXY
    \tkzFct[domain = 0.01:5, line width=1pt]{log(x)}
    \tkzText[draw,fill = brown!20](2,2){$f(x)=\ln(x)$}
\end{tikzpicture}

\begin{tikzpicture}[yscale=1.5, xscale=1]
    \tkzInit[xmin=-2,xmax=7,xstep=1,
    ymin=-2,ymax=2,ystep=1]
    \tkzGrid
    \tkzAxeXY
    \tkzFct[domain = -2:8, line width=1pt]{1 - exp(-x)}
    \tkzText[draw,fill = brown!20](1,1.5){$f(x)=1-e^{-x}$}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[yscale=.5, xscale=.8]
    \tkzInit[xmin=-5,xmax=5,xstep=1,
    ymin=-5,ymax=5,ystep=1]
    \tkzGrid
    \tkzAxeXY
    \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](-2,2){$f(x)=\frac{1}{x}$}
\end{tikzpicture}

\begin{tikzpicture}[yscale=0.5, xscale=.8]
    \tkzInit[xmin=-5,xmax=5,xstep=1,
    ymin=-1,ymax=10,ystep=1]
    \tkzGrid
    \tkzAxeXY
    \tkzFct[domain = -5:-0.01, line width=1pt]{1/x**2}
    \tkzFct[domain = 0.01:5, line width=1pt]{1/x**2}
    \tkzText[draw,fill = brown!20](3,3){$f(x)=\frac{1}{x^2}$}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[yscale=1.5, xscale=.8]
    \tkzInit[xmin=-5,xmax=5,xstep=1,
    ymin=-2,ymax=2,ystep=1]
    \tkzGrid
    \tkzAxeXY
    \tkzFct[domain = -5:5, line width=1pt]{cos(x)}
    \tkzText[draw,fill = brown!20](3,1){$f(x)=\cos{x}$}
\end{tikzpicture}

À l'aide des graphiques ci-dessus, déterminer graphiquement les quantités suivantes

\begin{multicols}{3}
    \begin{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow +\infty} x^2 = $
                \item $\ds \lim_{x\rightarrow -\infty} x^2 = $
            \end{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow +\infty} x^3 = $
                \item $\ds \lim_{x\rightarrow -\infty} x^3 = $
            \end{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow +\infty} e^x = $
                \item $\ds \lim_{x\rightarrow -\infty} e^x = $
            \end{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow +\infty} \ln(x) = $
                \item $\ds \lim_{x\rightarrow 0} \ln(x) = $
            \end{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow +\infty} 1-e^{-x} = $
                \item $\ds \lim_{x\rightarrow -\infty} 1-e^{-x} = $
            \end{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow -\infty} \frac{1}{x} = $
                \item $\ds \lim_{\substack{x\rightarrow 0 \\ <}} \frac{1}{x} = $
                \item $\ds \lim_{\substack{x\rightarrow 0 \\ >}} \frac{1}{x} = $
                \item $\ds \lim_{x\rightarrow +\infty} \frac{1}{x} = $
            \end{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow -\infty} \frac{1}{x^2} = $
                \item $\ds \lim_{\substack{x\rightarrow 0 \\ <}} \frac{1}{x^2} = $
                \item $\ds \lim_{\substack{x\rightarrow 0 \\ >}} \frac{1}{x^2} = $
                \item $\ds \lim_{x\rightarrow +\infty} \frac{1}{x^2} = $
            \end{enumerate}
        \item 
            \begin{enumerate}
                \item $\ds \lim_{x\rightarrow +\infty} \cos(x) = $
                \item $\ds \lim_{x\rightarrow -\infty} \cos(x) = $
            \end{enumerate}
    \end{enumerate}
\end{multicols}
\end{exercise}

\collectexercisesstop{banque}