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\begin{document}


\noindent {\LARGE PHEC1\hfill feuille d'exercices 9\hfill 2002-2003}

\noindent \hrulefill {\LARGE \bigskip }

\begin{exercise}[http://abdellah.bechata.free.fr]
D\'{e}terminer les limites suivantes%
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\begin{enumerate}
\item $\underset{x\rightarrow +\infty }{\lim }\dfrac{x^{3}+x^{2}+x\sqrt{x}%
+\ln x+\frac{1}{x}}{e^{3x}+e^{2x}+e^{x}+1+e^{-2x}}$

\item $\underset{x\rightarrow 0^{+}}{\lim }\dfrac{x^{3}+x^{2}+x\sqrt{x}+\ln
x+\frac{1}{x}}{e^{3x}+e^{2x}+e^{x}+1+e^{-2x}}$

\item $\underset{x\rightarrow +\infty }{\lim }\sqrt{x^{2}+x+1}-x+2$

\item $\underset{x\rightarrow -\infty }{\lim }\sqrt{x^{2}+x+1}-x+2$

\item $\underset{x\rightarrow 1^{+}}{\lim }\exp (\dfrac{1}{1-x})$

\item $\underset{x\rightarrow 1^{-}}{\lim }\exp (\dfrac{1}{1-x})$

\item $\underset{x\rightarrow 0}{\lim }\dfrac{\sin 3x}{2x}$

\item $\underset{x\rightarrow 0}{\lim }\dfrac{\ln (1+x^{2})}{x}$

\item $\underset{x\rightarrow +\infty }{\lim }(1+\frac{1}{x})^{x}$

\item $\underset{x\rightarrow -\infty }{\lim }(1+\frac{1}{x})^{x}$

\item $\underset{x\rightarrow 0^{+}}{\lim }(1+\frac{1}{x})^{x}$

\item $\underset{x\rightarrow 0^{-}}{\lim }(1+\frac{1}{x})^{x}$
\end{enumerate}

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\end{exercise}

\begin{exercise}
Compl\'{e}ter les \'{e}quivalents suivant

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\begin{enumerate}
\item $\sqrt{e^{2x}+e^{x}+1}-e^{x}\underset{x\rightarrow +\infty }{\sim }$

\item $\sqrt{e^{2x}+e^{x}+1}-e^{x}\underset{x\rightarrow -\infty }{\sim }$

\item $\ln (e^{x}+e^{-x})\underset{x\rightarrow +\infty }{\sim }$

\item $x\ln (1+\frac{1}{x})\underset{x\rightarrow +\infty }{\sim }$

\item $x\ln (1+\frac{1}{\sqrt{x}})\underset{x\rightarrow 0^{+}}{\sim }$

\item $\dfrac{1-\cos (x^{2})}{3x}\underset{x\rightarrow 0}{\sim }$
\end{enumerate}

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\begin{exercise}
D\'{e}terminer les d\'{e}veloppements limit\'{e}s \`{a} l'ordre $2$ en $0$
des fonctions suivantes :%
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\begin{enumerate}
\item $a(x)=e^{-x}\dfrac{x}{1+x}$

\item $b(x)=\dfrac{\sqrt{1-x}-\sqrt[3]{1+2x}}{x}$

\item $c(x)=\dfrac{e^{x}-1}{x}$

\item $e(x)=\ln (\cos 2x)$

\item $g(x)=\dfrac{\sin x-x}{x^{3}}$

\item $i(x)=\dfrac{\ln (1+x^{2})}{x}$
\end{enumerate}

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\begin{exercise}
D\'{e}terminer les limites suivantes%
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\begin{enumerate}
\item $\underset{x\rightarrow 0}{\lim }\dfrac{xe^{x}-e^{x}+1}{x^{2}}.$

\item $\underset{x\rightarrow 0}{\lim }\dfrac{x\cos x-\sin x}{x^{3}}$

\item $\underset{x\rightarrow +\infty }{\lim }x^{2}\ln (1+\dfrac{1}{x\sqrt{x}%
})$

\item $\underset{x\rightarrow 0}{\lim }\dfrac{\sqrt{1+x}-\sqrt{1-x}}{e^{x}-1}
$

\item $\underset{x\rightarrow 0}{\lim }\dfrac{x-(1+x)\ln (1+x)}{x^{2}}$

\item $\underset{x\rightarrow 0}{\lim }\dfrac{x\ln (1+x)-\sin x}{1-\cos x}$
\end{enumerate}

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\begin{exercise}
D\'{e}terminer le domaine de d\'{e}finition des fonctions suivantes et
expliciter leurs asymptotes aux bornes de leurs domaines de d\'{e}finition
respectives.%
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\begin{enumerate}
\item $a(x)=x+\sqrt{x^{2}-3x+2}$

\item $b(x)=\ln (e^{2x}+e^{x}+1)$

\item $c(x)=\ln (\dfrac{e^{2x}+1}{e^{x}+1})$

\item $d(x)=\ln (2^{x}+3^{x})$

\item $e(x)=\dfrac{x^{2}+x+1}{\left\vert x\right\vert +2}$

\item $f(x)=(1+\dfrac{1}{x^{2}})^{x}$
\end{enumerate}

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