We consider the function $f$ defined on $[0; +\infty[$ by
$$f(x) = \ln\left(\frac{3x+1}{x+1}\right).$$
We admit that the function $f$ is differentiable on $[0; +\infty[$ and we denote by $f'$ its derivative function. We denote by $\mathscr{C}_f$ the representative curve of the function $f$ in an orthogonal coordinate system.

\section*{Part A}
\begin{enumerate}
  \item Determine $\lim_{x \rightarrow +\infty} f(x)$ and give a graphical interpretation.
  \item a. Prove that, for every non-negative real number $x$,
$$f'(x) = \frac{2}{(x+1)(3x+1)}$$
b. Deduce that the function $f$ is strictly increasing on $[0; +\infty[$.
\end{enumerate}

\section*{Part B}
Let $(u_n)$ be the sequence defined by
$$u_0 = 3 \text{ and, for every natural number } n,\ u_{n+1} = f(u_n).$$
\begin{enumerate}
  \item Prove by induction that, for every natural number $n$, $\frac{1}{2} \leqslant u_{n+1} \leqslant u_n$.
  \item Prove that the sequence $(u_n)$ converges to a strictly positive limit.
\end{enumerate}

\section*{Part C}
We denote by $\ell$ the limit of the sequence $(u_n)$. We admit that $f(\ell) = \ell$.\\
The objective of this part is to determine an approximate value of $\ell$.\\
We introduce for this purpose the function $g$ defined on $[0; +\infty[$ by $g(x) = f(x) - x$.\\
We give below the table of variations of the function $g$ on $[0; +\infty[$ where $x_0 = \frac{-2+\sqrt{7}}{3} \approx 0.215$ and $g(x_0) \approx 0.088$, rounded to $10^{-3}$.

\begin{center}
\begin{tabular}{ | c | l c c | }
\hline
$x$ & 0 & $x_0$ & $+\infty$ \\
\hline
Variations &  & $g(x_0)$ &  \\
of the &  &  &  \\
function $g$ & 0 &  & $-\infty$ \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item Prove that the equation $g(x) = 0$ has a unique strictly positive solution. We denote it by $\alpha$.
  \item a. Copy and complete the algorithm below so that the last value taken by the variable $x$ is an approximate value of $\alpha$ by excess to 0.01 near.\\
b. Give then the last value taken by the variable $x$ during the execution of the algorithm.
$$x \leftarrow 0.22$$
While $\_\_\_\_$ do
$$x \leftarrow x + 0.01$$
End While
  \item Deduce an approximate value to 0.01 near of the limit $\ell$ of the sequence $(u_n)$.
\end{enumerate}