For every natural number $n$, the function $f_{n}$ is defined for all real $x$ in the interval $[0; 1]$ by:
$$f_{n}(x) = x + \mathrm{e}^{n(x-1)}$$
Let $\mathscr{C}_{n}$ denote the graph of the function $f_{n}$ in an orthogonal coordinate system.
Part A: generalities on the functions $\boldsymbol{f}_{\boldsymbol{n}}$
- Prove that, for every natural number $n$, the function $f_{n}$ is increasing and positive on the interval $[0; 1]$.
- Show that the curves $\mathscr{C}_{n}$ all have a common point A, and specify its coordinates.
- Using the graphical representations, can one conjecture the behavior of the slopes of the tangent lines at A to the curves $\mathscr{C}_{n}$ for large values of $n$? Prove this conjecture.
Part B: evolution of $\boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})$ when $x$ is fixed
Let $x$ be a fixed real number in the interval $[0; 1]$. For every natural number $n$, we set $u_{n} = f_{n}(x)$.
- In this question, assume that $x = 1$. Study the possible limit of the sequence $(u_{n})$.
- In this question, assume that $0 \leq x < 1$. Study the possible limit of the sequence $(u_{n})$.
Part C: area under the curves $\mathscr{C}_{\boldsymbol{n}}$
For every natural number $n$, let $A_{n}$ denote the area, expressed in square units, of the region located between the $x$-axis, the curve $\mathscr{C}_{n}$ and the lines with equations $x = 0$ and $x = 1$ respectively. Based on the graphical representations, conjecture the limit of the sequence $(A_{n})$ as the integer $n$ tends to $+\infty$, then prove this conjecture.