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%%%     Feuille d'exercices de mathématiques au format LaTeX     %%%
%%%     créée le Tue, 20 Jan 2009 10:13:06 +0100                 %%%
%%%     sur http://allken-bernard.org/pierre/phpmyexercices      %%%
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% Apparence des exercices
\newcounter{numexercice}
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\lhead{\textbf{\today}} 
\chead{\textbf{\thetitle}}
\rhead{\textbf{\thepage/\pageref{fin}}}
\lfoot{\small \textit{Lycée Joachim du Bellay}}
\cfoot{\small \textit{Mathématiques, prépa ECE1}}
\rfoot{\small \textit{http://allken-bernard.org/pierre/ece}}
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%%%     Exercice 439
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer les limites suivantes~:
\begin{eqnarray*}
A&=&\lim_{x\to +\infty} x^2-x\\
B&=&\lim_{x\rightarrow -\infty} x^{2009}-x^{2008}\\
C&=&\lim_{x\rightarrow +\infty} \frac{x-1}{x^3-x^2-x+1}\\
D&=&\lim_{x\rightarrow -\infty} \frac{x-1}{x^3-x^2-x+1}\\
E&=&\lim_{x\rightarrow 1_-} \frac{x-1}{x^3-x^2-x+1}\\
F&=&\lim_{x\rightarrow 1_+} \frac{x-1}{x^3-x^2-x+1}\\
G&=&\lim_{x\rightarrow +\infty} \frac{2x^2-x-1}{2x+1}\\
H&=&\lim_{x\rightarrow -\infty} \frac{2x^2-x-1}{2x+1}\\
I&=&\lim_{x\rightarrow -\frac{1}{2}_-} \frac{2x^2-x-1}{2x+1}\\
J&=&\lim_{x\rightarrow -\frac{1}{2}_+} \frac{2x^2-x-1}{2x+1}\\
K&=&\lim_{x\rightarrow +\infty} \frac{\frac{x-1}{x+2}-\frac{x+1}{x-2}}{\frac{x+3}{x-4}-\frac{x-3}{x+4}}\\
L&=&\lim_{x\rightarrow 3} \frac{x^3-27}{x-3}\\
M&=&\lim_{x\rightarrow -1_-} \frac{x+1}{x^3+4x^2+5x+2}\\
N&=&\lim_{x\rightarrow -1_+} \frac{x+1}{x^3+4x^2+5x+2}\\
O&=&\lim_{x\rightarrow 1} \frac{x^n-1}{x-1}\quad (n\in\mathbf N^*)\\
\end{eqnarray*}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 440
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer les limites suivantes~:
\begin{eqnarray*}
A&=&\lim_{x\rightarrow +\infty} \frac{x^2-\sqrt{x}}{\sqrt{x}-x}\\
B&=&\lim_{x\rightarrow +\infty} \frac{3\sqrt{x}-x^2+x}{2x^3-\sqrt x}\\
C&=&\lim_{x\rightarrow 2} \frac{\sqrt{x+2}-2}{x-2}\\
D&=&\lim_{x\rightarrow +\infty} \sqrt{x^2+1}-x\\
E&=&\lim_{x\rightarrow +\infty} \sqrt{2x^2+x+1}-{\sqrt 2}\,x\\
F&=&\lim_{x\rightarrow -\infty} \sqrt{2x^2+x+1}-{\sqrt 2}\,x\\
G&=&\lim_{x\rightarrow -\infty} \frac{\sqrt{x+1}-\sqrt{x}}{\sqrt{x+2}-\sqrt{x}}\\
H&=&\lim_{x\rightarrow 2} \frac{\sqrt{2x}-2}{\sqrt{x+7}-3}\\
I&=&\lim_{x\rightarrow +\infty} \left(\frac{x}{2}-\frac{\sqrt{x^2-1}}{x}\right)\\
J&=&\lim_{x\rightarrow +\infty} \frac{1}{x}\left(\frac{x}{2}-\frac{\sqrt{x^2-1}}{x}\right)\\
K&=&\lim_{h\rightarrow 0} \frac{\sqrt{1+h}-1}{h}\\
L&=&\lim_{x\rightarrow +\infty} \sqrt[3]{x^3+1}-x\\
\end{eqnarray*}
\end{exercice}


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%%%     Exercice 442
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer les limites suivantes~:
\begin{eqnarray*}
A&=&\lim_{x\to +\infty} \frac{e^x}{x}\\
B&=&\lim_{x\to +\infty} \frac{x}{e^x}\\
C&=&\lim_{x\to +\infty} \frac{e^x+1}{x}\\
D&=&\lim_{x\to +\infty} \frac{e^x}{x+1}\\
E&=&\lim_{x\to +\infty} \frac{x^2e^x}{x^3+x+1}\\
F&=&\lim_{x\to +\infty} \frac{x^2e^x}{x e^{2x}-1}\\
G&=&\lim_{x\to +\infty} e^x-x^4\\
H&=&\lim_{x\to +\infty} \frac{e^x-x}{\sqrt x}\\
I&=&\lim_{x\to +\infty} x^3-2\sqrt x+e^x\\
J&=&\lim_{x\to +\infty} e^{2x}-x\\
K&=&\lim_{x\to +\infty} \frac{e^{x^2}+1}{x+1}\\
L&=&\lim_{x\to -\infty} xe^x\\
M&=&\lim_{x\to -\infty} e^x-x\\
N&=&\lim_{x\to 0_+} xe^{\frac{1}{x}}\\
O&=&\lim_{x\to +\infty} \frac{e^{e^x}}{e^x(e^x-x)}\\
\end{eqnarray*}
\end{exercice}

\begin{exercice}
On pose~:
\[f(x)=\frac{\sqrt{1+x}-1}{x}\]
\begin{itemize}
\item Déterminer le domaine de définition de $f$.
\item Montrer que $f$ peut-être prolongée par continuité en $0$.
\end{itemize}
\end{exercice}

\begin{exercice}
On pose~:
\[f(x)=\frac{\ln(1-x)}{x}\]
\begin{itemize}
\item Déterminer le domaine de définition de $f$.
\item Montrer que $f$ peut-être prolongée par continuité en $0$.
\end{itemize}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 443
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer les limites suivantes~:
\begin{eqnarray*}
A&=&\lim_{x\to 0} \frac{e^x-1}{x}\\
B&=&\lim_{x\to 0} \frac{\ln(1+x)}{x}\\
C&=&\lim_{x\to +\infty} x(e^{\frac{1}{x}}-1)\\
D&=&\lim_{x\to +\infty} x(\ln(x+1)-\ln(x))\\
E&=&\lim_{x\to 0} \frac{2^x-1}{x}\\
F&=&\lim_{x\to 1} \frac{\ln(x)}{x-1}\\
G&=&\lim_{x\to 0} \frac{e^x-1}{\ln(1+x)}\\
H&=&\lim_{x\to 0} \frac{e^x-1}{\sqrt{x}}\\
I&=&\lim_{x\to 0} \frac{e^x-1}{\sqrt{1+x}-1}\\
J&=&\lim_{x\to 0} \frac{\ln(1+x)}{(1+x)^4-1}\\
K&=&\lim_{x\to 0} \frac{\ln(1+x)}{x^2}\\
\end{eqnarray*}
\end{exercice}


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%%%     Exercice 446
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer les limites suivantes~:
\begin{eqnarray*}
A&=&\lim_{x\rightarrow 0_+} \frac{x}{7}\left\lfloor \frac{3}{x}\right\rfloor\\
B&=&\lim_{x\rightarrow 0_-} \frac{x}{7}\left\lfloor \frac{3}{x}\right\rfloor\\
C&=&\lim_{x\rightarrow +\infty} \frac{x}{7}\left\lfloor \frac{3}{x}\right\rfloor\\
D&=&\lim_{x\rightarrow -\infty} \frac{x}{7}\left\lfloor \frac{3}{x}\right\rfloor\\
\end{eqnarray*}
Indice~: utiliser des encadrements.
\end{exercice}


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%%%     Exercice 444
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Etudier la convergence des suites suivantes~:
\begin{eqnarray*}
a_n&=&\frac{n}{n^2+1}\\
b_n&=&\sqrt{n^4+1}-\sqrt{n^4-1}\\
c_n&=&\frac{1}{n^2 e^{-n}}\\
d_n&=& n\ln\left(1+\frac{1}{n}\right)\\
e_n&=& n(e^{\frac{2}{n}+1}-e)\\
f_n&=&\left (\frac{1}{3}\right)^n\\
g_n&=&\left (-\frac{1}{3}\right)^n\\
h_n&=&3^n\\
i_n&=&(-3)^n\\
j_n&=&n\left(\frac{1}{2}\right)^n\\
k_n&=&\frac{n^2+3n-4}{(n-1)\left(\frac{1}{5}\right)^n}\\
\end{eqnarray*}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 445
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Donner un équivalent simple de chaque suite.
\begin{eqnarray*}
a_n&=&n^3+n^2+n+1\\
b_n&=&\sqrt n -n\\
c_n&=&\frac{2n+1}{n^2-1}\\
d_n&=&\frac{e^n-\sqrt n}{n^2e^n+n+1}\\
e_n&=&e^{\frac{1}{n}}-1\\
f_n&=&(n^2-2n+2)(e^{\frac{1}{n^2}}-1)\\
g_n&=&\ln\left(1+\frac{1}{n}\right)\\
h_n&=&(n^3+n+1)(e^{\frac{1}{\sqrt{n}}}-1)\ln(1+n^{-2})\\
i_n&=&\frac{n^2\ln(1+e^{-n})}{n+1}\\
j_n&=&\left(\frac{1}{2}\right)^n+\left(\frac{1}{3}\right)^n+\left(\frac{1}{4}\right)^n\\
k_n&=&\alpha^n+\beta^n\quad\textrm{avec $|\alpha|<|\beta|<1$}\\
\end{eqnarray*}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 441
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer les limites suivantes~:
\begin{eqnarray*}
A&=&\lim_{x\to +\infty} e^{2x+1}-x^5\\
B&=&\lim_{x\to +\infty} \frac{e^{3x}-x}{\ln(x)}\\
C&=&\lim_{x\to +\infty} e^{-{x^2}}-\ln(x)\\
D&=&\lim_{x\to 0} \frac{\ln(1+x)}{x^2}\\
E&=&\lim_{x\to 0_+} \frac{\sqrt x}{e^x-1}\\
F&=&\lim_{x\to 0_+} \frac{2\sqrt x}{\ln(1+x)}\\
G&=&\lim_{x\to 0} \frac{e^x-1}{\ln(1+x)}\\
H&=&\lim_{x\to +\infty} x^4 e^{-\sqrt x}\\
I&=&\lim_{x\to +\infty} x\ln\left(1+\frac{1}{x}\right)\\
J&=&\lim_{x\to 0_+} \frac{\ln(1+4x)}{x}\\
K&=&\lim_{x\to 0_+} \frac{\ln(1+x\sqrt x)}{x^2}\\
L&=&\lim_{x\to 0_+} \frac{x}{e^{x^2}-1}\\
M&=&\lim_{x\to 0_+} 3x^2+(\ln x)^2-\frac{1}{\sqrt x}\\
N&=&\lim_{x\to +\infty} \frac{x+x^2\ln(x)}{\ln(x)^2+\ln(x^2)}\\
O&=&\lim_{x\to 0_+} (1+x^2)^x\\
P&=&\lim_{x\to +\infty} \left(1-\frac{2}{x}\right)^x\\
Q&=&\lim_{x\to +\infty} \left(1+\frac{1}{x^2}\right)^{\sqrt x}\\
R&=&\lim_{x\to -\infty} (2^x+3^x)^{1/x}\\
S&=&\lim_{x\to +\infty} \frac{\sqrt{x^4+x+1}}{x^2+1}\\
T&=&\lim_{x\to +\infty} (x+1)^{1/3}-x^{1/3}\\
U&=&\lim_{x\to +\infty} \exp(x^2)-e^{3x}+x^2\\
V&=&\lim_{x\to +\infty} \ln(x^2+1)-2\ln(x)\\
\end{eqnarray*}


\end{exercice}

\label{fin}
\end{document}