In the plane equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, we denote by $\mathscr{C}_{u}$ the curve representing the function $u$ defined on the interval $]0; +\infty[$ by:
$$u(x) = a + \frac{b}{x} + \frac{c}{x^2}$$
where $a, b$ and $c$ are fixed real numbers.

The curve $\mathscr{C}_{u}$ passes through the points $\mathrm{A}(1; 0)$ and $\mathrm{B}(4; 0)$ and the $y$-axis and the line $\mathscr{D}$ with equation $y = 1$ are asymptotes to the curve $\mathscr{C}_{u}$.

\begin{enumerate}
  \item Give the values of $u(1)$ and $u(4)$.
  \item Give $\lim_{x \rightarrow +\infty} u(x)$. Deduce the value of $a$.
  \item Deduce that, for all strictly positive real $x$, $u(x) = \frac{x^2 - 5x + 4}{x^2}$.
\end{enumerate}