\section*{Exercise 1}
\section*{Part A}
We consider the function $g$ defined on the interval $] 0 ; + \infty [$ by

$$g ( x ) = \ln \left( x ^ { 2 } \right) + x - 2$$

\begin{enumerate}
  \item Determine the limits of the function $g$ at the boundaries of its domain.
  \item It is admitted that the function $g$ is differentiable on the interval $] 0 ; + \infty [$. Study the variations of the function $g$ on the interval $] 0 ; + \infty [$.
  \item a. Prove that there exists a unique strictly positive real number $\alpha$ such that $g ( \alpha ) = 0$.\\
b. Determine an interval containing $\alpha$ with amplitude $10 ^ { - 2 }$.
  \item Deduce the sign table of the function $g$ on the interval $] 0 ; + \infty [$.
\end{enumerate}

\section*{Part B}
We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by :

$$f ( x ) = \frac { ( x - 2 ) } { x } \ln ( x ) .$$

We denote $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system.

\begin{enumerate}
  \item a. Determine the limit of the function $f$ at 0.\\
b. Interpret the result graphically.
  \item Determine the limit of the function $f$ at $+ \infty$.
  \item It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.
\end{enumerate}

Show that for every strictly positive real number $x$, we have $f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.\\
4. Deduce the variations of the function $f$ on the interval $] 0 ; + \infty [$.

\section*{Part C}
Study the relative position of the curve $\mathscr { C } _ { f }$ and the representative curve of the natural logarithm function on the interval $] 0 ; + \infty [$.