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\begin{document}
$\mathbf{1)}$

$i)\qquad$Calcolare%
\[
\lim_{x\rightarrow+\infty}\frac{\int_{2}^{x}\frac{1}{\log\left(  t\right)
}~dt}{\frac{x}{\log\left(  x\right)  }}\;.
\]


$ii)\qquad$Calcolare%
\[
\lim_{x\rightarrow+\infty}\left(  \int_{2}^{x}\frac{1}{\log\left(  t\right)
}~dt-\frac{x}{\log\left(  x\right)  }\right)  .
\]


$\bigskip$

$\mathbf{2)\qquad}$

$i)\qquad$Tracciare un grafico della funzione%
\[
h\left(  x\right)  =\frac{\log\left(  x\right)  }{x}\;,
\]
evidenziandone l'insieme di definizione, i limiti, il crescere e il
decrescere, la convessit\`{a} e la concavit\`{a}.

$ii)\qquad$Dedurre che%
\[
e^{\pi}>\pi^{e}\;.
\]


$\bigskip$

$\mathbf{3)}$

$i)\qquad$Calcolare%
\[
A=\lim_{R\rightarrow+\infty}\left(  \int_{R/2}^{2R}\frac{1}{x^{\alpha}%
}~dx\right)
\]
al variare del parametro reale $\alpha$.

$ii)\qquad$Calcolare%
\[
B=\lim_{N\rightarrow+\infty}\left(  \sum_{j=N+1}^{2N}\ \frac{1}{j}\right)  \;.
\]


\bigskip

$\mathbf{4)\qquad}$Sia%
\[
A=\left\{  \left(  x,y\right)  \in\mathbb{R}^{2}:x\geq0,\ \ 0\leq y\leq
e^{-x}\right\}  \ .
\]
Determinare, tra tutti i triangoli rettangoli contenuti in $A$ e aventi i
cateti inclusi negli assi, quello di area massima.


\end{document}