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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N� 13156991010852--
%%--Limites de fonctions--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%
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\label{Force2pass}

\begin{exercise}
Etudier la limite de $f(x)$ en $+\infty$, \`{a} l'aide des op\'{e}rations sur
les limites:%

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\begin{enumerate}
\item $f(x)=3x^{3}+\sqrt{x}.$

\item $f(x)=2+\frac{1}{2+\frac{1}{x}}.$

\item $f(x)=3x^{2}-x+1$\quad\textit{(mettre }$x^{2}$\textit{\ en facteur)}.

\item $f(x)=\frac{x^{2}-1}{2x^{3}+1}$\quad\textit{(\'{e}crire }$f(x)=\frac
{x^{2}(...)}{x^{3}(...)}$\textit{)}.
\end{enumerate}%

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\end{exercise}

\begin{exercise}
A l'aide des th\'{e}or\`{e}mes de comparaison \'{e}tudier la limite de:%

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\begin{enumerate}
\item $f(x)=\cos x-3x\quad$en $+\infty$.

\item $f(x)=\frac{\sin x}{\sqrt{x}}\quad$en $+\infty$.

\item $f(x)=\frac{x\sin x}{x^{2}+1}\quad$en $+\infty$.

\item $f(x)=x\sin(\frac{1}{x})\quad$en $0$.
\end{enumerate}%

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\end{exercise}

\begin{exercise}
En utilisant la limite d'une fonction compos\'{e}e \'{e}tudier la limite de:%

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\begin{enumerate}
\item $f(x)=\sqrt{x^{2}-1}\quad$en $+\infty$.

\item $f(x)=\sqrt{\frac{x+3}{x-3}}$\quad en $+\infty$.

\item $f(x)=\frac{\sin(\sqrt{x})}{\sqrt{x}}\quad$\ en $0^{+}$

\textit{(effectuer le changement de variable }$X=\sqrt{x}$\textit{)}.

\item $f(x)=\frac{\sin(3x)}{x}\quad$\ en $0$.
\end{enumerate}%

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\end{exercise}

\begin{exercise}
Etudier la limite de:%

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\begin{enumerate}
\item $f(x)=\frac{-8x^{2}+1}{5x^{2}+x+1}\quad$\ en $+\infty$.

\item $f(x)=\frac{x^{n}}{2x^{2}+1}\quad$\ en $+\infty$

\textit{(discuter suivant les valeurs de }$n\in%
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\mathbb{N}
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$\textit{)}.

\item $f(x)=x\sqrt{x}-5x^{2}\quad$\ en $+\infty$.

\item $f(x)=\frac{1-3x}{x-\sqrt{x}}\quad$\ en $+\infty$.

\item $f(x)=\frac{\sqrt{x}-3}{x-9}\quad$\ en $9$.

\item $f(x)=\frac{\sqrt{x+2}-2}{\sqrt{x+7}-3}\quad$\ en $2$.

\item $f(x)=\sqrt{x^{2}+4x+3}-x\quad$\ en $+\infty$.

\item $f(x)=x+\sqrt{x^{2}+1}\quad$\ en $-\infty$.
\end{enumerate}%

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\end{exercise}

\begin{exercise}
Dans la th\'{e}orie de la relativit\'{e}, Einstein a d\'{e}montr\'{e} que la
longueur $L$ et la masse $m$ d'un objet en mouvement
\textbf{d\'{e}pendent~de~sa~vitesse~}$v$:%
\[
L(v)=L_{0}\sqrt{1-\frac{v^{2}}{c^{2}}}\quad\text{et\quad}m(v)=\frac{m_{0}%
}{\sqrt{1-\frac{v^{2}}{c^{2}}}}%
\]
o\`{u} $L_{0}$ et $m_{0}$ sont, respectivement, la longueur et la masse de
l'objet sans vitesse; $c$ est la vitesse de la lumi\`{e}re.

\begin{enumerate}
\item Combien mesureriez-vous si vous vous d\'{e}placiez \`{a} la moiti\`{e}
de la vitesse de la lumi\`{e}re.

\item D\'{e}terminer $\underset{v\rightarrow c^{-}}{\lim}L(v)$ et
$\underset{v\rightarrow c^{-}}{\lim}m(v)$. Interpr\'{e}ter physiquement ces r\'{e}sultats.
\end{enumerate}
\end{exercise}


\end{document}