\textbf{Exercise 4 — Theme: Functions, Exponential Function, Logarithm Function; Sequences}

\textbf{Part A}\\
Consider the function $f$ defined for every real $x$ in $]0; 1]$ by:
$$f(x) = \mathrm{e}^{-x} + \ln(x).$$
\begin{enumerate}
  \item Calculate the limit of $f$ at 0.
  \item It is admitted that $f$ is differentiable on $]0; 1]$. Let $f'$ denote its derivative function.\\
Prove that, for every real $x$ belonging to $]0; 1]$, we have:
$$f'(x) = \frac{1 - x\mathrm{e}^{-x}}{x}$$
  \item Justify that, for every real $x$ belonging to $]0; 1]$, we have $x\mathrm{e}^{-x} < 1$.\\
Deduce the variation table of $f$ on $]0; 1]$.
  \item Prove that there exists a unique real $\ell$ belonging to $]0; 1]$ such that $f(\ell) = 0$.
\end{enumerate}

\textbf{Part B}
\begin{enumerate}
  \item Two sequences $(a_n)$ and $(b_n)$ are defined by:
$$\left\{\begin{array}{l} a_0 = \frac{1}{10} \\ b_0 = 1 \end{array}\right. \text{ and, for every natural number } n, \left\{\begin{array}{l} a_{n+1} = \mathrm{e}^{-b_n} \\ b_{n+1} = \mathrm{e}^{-a_n} \end{array}\right.$$
a. Calculate $a_1$ and $b_1$. Approximate values to $10^{-2}$ will be given.\\
b. Consider below the function terms, written in Python language.
\begin{verbatim}
def termes (n) :
    a=1/10
    b=1
    for k in range(0,n) :
        c= ...
        b = ...
        a = c
    return(a,b)
\end{verbatim}
Copy and complete without justification the box above so that the function termes calculates the terms of the sequences $(a_n)$ and $(b_n)$.
  \item Recall that the function $x \longmapsto \mathrm{e}^{-x}$ is decreasing on $\mathbb{R}$.\\
a. Prove by induction that, for every natural number $n$, we have:
$$0 < a_n \leqslant a_{n+1} \leqslant b_{n+1} \leqslant b_n \leqslant 1$$
b. Deduce that the sequences $(a_n)$ and $(b_n)$ are convergent.
  \item Let $A$ denote the limit of $(a_n)$ and $B$ denote the limit of $(b_n)$.\\
It is admitted that $A$ and $B$ belong to the interval $]0; 1]$, and that $A = \mathrm{e}^{-B}$ and $B = \mathrm{e}^{-A}$.\\
a. Prove that $f(A) = 0$.\\
b. Determine $A - B$.
\end{enumerate}