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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1315705456654--
%%--Propriétés caractéristiques d\'une fonction admettant une asymptote oblique--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
Dans un rep\'{e}re orthogonal du plan, on note $\mathcal{C}_{f}$ la courbe
repr\'{e}sentative d'une fonction $f$ d\'{e}finie au voisinage de $+\infty$ et
telle que $\underset{x\rightarrow+\infty}{\lim}f(x)=+\infty$.

\begin{enumerate}
\item Supposons que la droite $\left(  \Delta\right)  $ d'\'{e}quation
$y=ax+b\quad(a\neq0)$ soit asymptote \`{a} $\mathcal{C}_{f}$.

Posons alors $g(x)=f(x)-(ax+b)$; on sait donc que $\underset{x\rightarrow
+\infty}{\lim}g(x)=0$.

\begin{enumerate}
\item D\'{e}montrer que $a=\underset{x\rightarrow+\infty}{\lim}\frac{f(x)}{x}$.

\item D\'{e}montrer que $b=\underset{x\rightarrow+\infty}{\lim}\left(
f(x)-ax\right)  $.
\end{enumerate}

\item R\'{e}ciproquement, on suppose qu'il existe deux r\'{e}els $a$ et $b$
($a\neq0$) tels que:%
\[
a=\underset{x\rightarrow+\infty}{\lim}\frac{f(x)}{x}\text{\quad et\quad
}b=\underset{x\rightarrow+\infty}{\lim}\left(  f(x)-ax\right)  .
\]


D\'{e}montrer que la droite $\left(  \Delta\right)  $ d'\'{e}quation $y=ax+b$
est asymptote \`{a} $\mathcal{C}_{f}.$

\item Synth\'{e}tiser, en une phrase, les r\'{e}sultats d\'{e}montr\'{e}s.

\item \textit{Application:} montrer que la courbe d'\'{e}quation
$y=\frac{x^{2}+5}{2x+1}$ admet en $+\infty$ une asymptote oblique dont on
d\'{e}terminera une \'{e}quation cart\'{e}sienne.
\end{enumerate}

\textit{Remarque: le jour du BAC, une \'{e}quation de l'asymptote est, en
g\'{e}n\'{e}ral, donn\'{e}e.}
\end{exercise}


\end{document}