We consider the function $f$ defined on $\mathbb { R }$ by:

$$f ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$$

The representative curve $\mathscr { C }$ of the function $f$ in an orthonormal coordinate system is given.

\begin{enumerate}
  \item Calculate the limit of the function $f$ at negative infinity and interpret the result graphically.
  \item Show that the line with equation $y = 2$ is a horizontal asymptote to the curve $\mathscr { C }$.
  \item Calculate $f ^ { \prime } ( x )$, where $f ^ { \prime }$ is the derivative function of $f$, and verify that for all real numbers $x$ we have:
$$f ^ { \prime } ( x ) = \frac { f ( x ) } { \mathrm { e } ^ { x } + 1 } .$$
  \item Show that the function $f$ is increasing on $\mathbb { R }$.
  \item Show that the curve $\mathscr { C }$ passes through the point $\mathrm { I } ( 0 ; 1 )$ and that its tangent at this point has slope 0.5.
\end{enumerate}