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\begin{document}

 Maths cc z~~~~~~~~~~~~~~~~{\Huge {\bf 
                      ~Exo-type 7}}  \hfill x/x/x ~Deug MIAS\\ %

		     \vspace{0.6cm}
\noindent
Nom:\\
Documents:\\
Calculette:
		     \vspace{0.6cm}


Dans ce qui suit, $n$ est entier, tandis que $x$ et $y$ sont r\'eels; $n$ et $x$
tendent vers $+\infty$, tandis que y tend vers $0$.
Pour chacune des fonctions et  des suites num\'eriques \'evoqu\'ees ci-dessous, donner 
un \'equivalent aussi simple que possible. 

$$
\begin{array}{|c|c|}
\hline
a_n = 2^n -4n &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
~&~\\ \hline
b_n = \ln (3n -8) - \ln (4n -2)&\\ ~&~\\ \hline
c_n = {a_n}^{b_n}&\\ ~&~\\ \hline
f(x)= xe^{-x}+3x^2+2x^3&\\ ~&~\\ \hline
g(y)= ye^{-y}+3y^2+2y^3&\\ ~&~\\ \hline
h(x) = \sin x + x^3&\\ ~&~\\ \hline
k(y) = \sin y -y^3&\\ ~&~\\ \hline
m(x)=\ln \frac {\sqrt {9x^2+16} +\sqrt {16x^2+9}}{7x+5+\cos x}&\\ ~&~\\ \hline
p(y)=\ln \frac {\sqrt {9y^2+16} +\sqrt {16y^2+9}}{5y+7\cos y}&\\ ~&~\\ \hline
q(x)= e^{ \sqrt {x^2+ x}}&\\ ~&~\\ \hline
 r(x)= e^{ x^2-\sqrt {x^2+x}}&\\ ~&~\\ \hline
 s_n= e\,n \ln \left (1+\frac {1}{n}\right )- \left (1 + \frac {1}{n}
\right )^n &\\ ~&~\\ \hline
t_n= \tan \frac {1}{n} + n \sin ^2\frac {1}{n}&\\ ~&~\\ \hline
u_n = \frac {s_n}{t_n}&\\ ~&~\\ \hline
\end {array}
    $$

	\end{document}

$$ f(x)= xe^{-x}+3x^2+2x^3;\, ~~~~g(y)= ye^{-y}+3y^2+2y^3;$$

$$ h(x) = \sin x + x^3 ;~~~~
k(y) = \sin y -y^3;
$$

$$ m(x)=\ln \frac {\sqrt {9x^2+16} +\sqrt {16x^2+9}}{7x+5\cos x} ; ~~~~
p(y)=\ln \frac {\sqrt {9y^2+16} +\sqrt {16y^2+9}}{5y+7\cos y} ;$$

$$ q(x)= e^{ \sqrt {x^2+ x}}; ~~~~
r(x)= e^{ x^2-\sqrt {x^2+x}}; $$

et le bouquet:

$$ s_n= e\,n \ln \left (1+\frac {1}{n}\right )- \left (1 + \frac {1}{n}
\right )^n ;~~~~
t_n= \tan \frac {1}{n} + n \sin ^2\frac {1}{n} ; ~~~~u_n = \frac {s_n}{t_n}.$$

Donner pour chacune un \'equivalent aussi simple que possible.



	\end{document}
