\section*{Part I}
We consider the function $h$ defined on the interval $] 0 ; + \infty [$ by:
$$h ( x ) = 1 + \frac { \ln ( x ) } { x }$$

\begin{enumerate}
  \item Determine the limit of the function $h$ at 0.
  \item Determine the limit of the function $h$ at $+ \infty$.
  \item We denote $h ^ { \prime }$ the derivative function of $h$. Prove that, for every real number $x$ in $] 0 ; + \infty [$, we have:
$$h ^ { \prime } ( x ) = \frac { 1 - \ln ( x ) } { x ^ { 2 } }$$
  \item Draw up the variation table of the function $h$ on the interval $] 0 ; + \infty [$.
  \item Prove that the equation $h ( x ) = 0$ has a unique solution $\alpha$ in $] 0 ; + \infty [$. Justify that we have: $0.5 < \alpha < 0.6$.
\end{enumerate}

\section*{Part II}
In this part, we consider the functions $f$ and $g$ defined on $] 0 ; + \infty [$ by:
$$f ( x ) = x \ln ( x ) - x ; \quad g ( x ) = \ln ( x )$$
We denote $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ the curves representing respectively the functions $f$ and $g$ in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath })$.\\
For every strictly positive real number $a$, we call:
\begin{itemize}
  \item $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at its point with abscissa $a$;
  \item $D _ { a }$ the tangent to $\mathscr { C } _ { g }$ at its point with abscissa $a$.
\end{itemize}
We are looking for possible values of $a$ for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular.\\
Let $a$ be a real number belonging to the interval $] 0 ; + \infty [$.

\begin{enumerate}
  \item Justify that the line $D _ { a }$ has slope $\frac { 1 } { a }$.
  \item Justify that the line $T _ { a }$ has slope $\ln ( a )$.
  \item We recall that in an orthonormal coordinate system, two lines with slopes $m$ and $m ^ { \prime }$ respectively are perpendicular if and only if $m m ^ { \prime } = - 1$.\\
Prove that there exists a unique value of $a$, which you will identify, for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular.
\end{enumerate}