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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 13164384827627--
%%--Notion de limite--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%
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\label{ForcePass2}

\section{Un cas plut\^{o}t \'{e}vident}

On consid\`{e}re les fonctions $f$ et $g$ d\'{e}finies par:%
\[
\forall x\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\backslash\{2\};f(x)=\frac{x^{3}-x^{2}-x-2}{x-2}\quad\text{et\quad}\forall
x\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
;g(x)=x^{2}+x+1.
\]


\begin{enumerate}
\item A l'aide de la commande \fbox{Table} de votre calculatrice,
compl\'{e}ter le tableau suivant (\`{a} $10^{-3}$ pr\`{e}s):\medskip
\end{enumerate}

\begin{center}
$%
\begin{tabular}
[c]{|c|cc|cc|}\hline
$x$ &  & $f(x)$ &  & $g(x)$\\\hline
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{$1.7$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{$1.8$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{$1.9$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{$2$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{$2.1$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{$2.2$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace{4cm}}\\
$2.3$ &  & \hspace{4cm} &  & \hspace{4cm}\\\hline
\end{tabular}
$
\end{center}

\begin{enumerate}
\item[ ] Que remarque-t-on pour les valeurs de $x\neq2$ du tableau?

\item[\textbf{2)}] Que remarque-t-on si l'on trace les graphes de $f$ et $g$
sur le m\^{e}me \'{e}cran de la calculatrice?

Peut-on conclure que $f=g$? (Justifier)

\item[\textbf{3)}] 

\begin{enumerate}
\item Sans faire de calculs, compl\'{e}ter le tableau suivant :\medskip
\end{enumerate}
\end{enumerate}

\begin{center}
$%
\begin{tabular}
[c]{|c|cc|cc|}\hline
$x$ &  & $f(x)$ &  & $g(x)$\\\hline
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \hspace{4cm}\\
\multicolumn{1}{|l|}{$1.99$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} & $6.9501$\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \hspace{4cm}\\
\multicolumn{1}{|l|}{$1.999$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} & $6.995001$\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \hspace{4cm}\\
\multicolumn{1}{|l|}{$1.9999$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} & $6.99950001$\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \hspace{4cm}\\
\multicolumn{1}{|l|}{$2$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} & $7$\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \hspace{4cm}\\
\multicolumn{1}{|l|}{$2.01$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} & $7.0501$\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \hspace{4cm}\\
\multicolumn{1}{|l|}{$2.001$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}} & \multicolumn{1}{|l}{} & $7.005001$\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}} & \multicolumn{1}{|l}{} & \hspace{4cm}\\
$2.0001$ &  & \hspace{4cm} &  & $7.00050001$\\\hline
\end{tabular}
$\newpage
\end{center}

\begin{enumerate}
\item[\textbf{\ }] 

\begin{enumerate}
\item[\textbf{b)}] Compl\'{e}ter la phrase suivante:

$f(x)$ n'est pas definie en $x=\ldots\ldots$ mais on \textbf{constate} que
plus $x$ est proche de \ldots\ldots, plus $f(x)$ est proche de $g(\ldots
\ldots)=\ldots\ldots$.

\begin{definition}
On dit que la limite de $f(x)$ quand $x$ tend vers $2$ est \'{e}gale \`{a} $7
$ et l'on note cela par:%
\[
\underset{x\rightarrow2}{\lim}f(x)=\underset{x\rightarrow2}{\lim}\frac
{x^{3}-x^{2}-x-2}{x-2}=7\qquad\text{ou encore\qquad}f(x)\overset
{x\rightarrow2}{\rightarrow}7
\]

\end{definition}

\item[\textbf{c)}] Soit $P(x)=x^{3}-x^{2}-x-2$. D\'{e}montrer que $2$ est une
racine de $P$ et factoriser $P(x)$.

\item[\textbf{d)}] D\'{e}duire du c) une simplification de $f(x)$ pour
$x\neq2$.

\begin{remark}
On comprend, bien que $f$ ne soit pas d\'{e}finie en $2,$ que $f(x)$ se
rapproche de $g(2)=7$ quand $x$ se rapproche de $2.$
\end{remark}
\end{enumerate}
\end{enumerate}

\section{Un cas moins \'{e}vident}

On consid\`{e}re la fonction $f$ d\'{e}finie par:%
\[
\forall x\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{\star};\text{ }f(x)=\frac{\sin x}{x}.
\]


\begin{enumerate}
\item A l'aide de votre calculatrice, compl\'{e}ter le tableau suivant \`{a}
$10^{-9}$ pr\`{e}s \textbf{(v\'{e}rifier que vous \^{e}tes en mode
radian)}:\medskip
\end{enumerate}

\begin{center}
$%
\begin{tabular}
[c]{|c|cc|}\hline
$x$ &  & $f(x)$\\\hline
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}}\\
\multicolumn{1}{|l|}{$-0.1$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}}\\
\multicolumn{1}{|l|}{$-0.01$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}}\\
\multicolumn{1}{|l|}{$-0.001$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}}\\
\multicolumn{1}{|l|}{$0$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}}\\
\multicolumn{1}{|l|}{$0.1$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}}\\
\multicolumn{1}{|l|}{$0.01$} & \multicolumn{1}{|l}{} &
\multicolumn{1}{l|}{\hspace{4cm}}\\
\multicolumn{1}{|l|}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l|}{\hspace
{4cm}}\\
$0.001$ &  & \hspace{4cm}\\\hline
\end{tabular}
$
\end{center}

\begin{enumerate}
\item[\textbf{2)}] Conjecturer la limite de $f(x)$ quand $x$ tend vers $0$.

Connaissez-vous une simplification de $f(x)$ pour $x\neq0$?
\end{enumerate}

%

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