Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x - 5\ln x - \frac{4}{x}$$
Determine the limit of $f(x)$ as $x$ tends to 0. You may use without proof the fact that $\lim_{x \rightarrow 0} x \ln x = 0$.
Determine the limit of $f(x)$ as $x$ tends to $+\infty$.
Prove that, for all strictly positive real $x$, $f'(x) = u(x)$, where $u(x) = \frac{x^2 - 5x + 4}{x^2}$.
Deduce the table of variations of the function $f$ by specifying the limits and particular values.
Let $f$ be the function defined on the interval $]0; +\infty[$ by:
$$f(x) = x - 5\ln x - \frac{4}{x}$$
\begin{enumerate}
\item Determine the limit of $f(x)$ as $x$ tends to 0. You may use without proof the fact that $\lim_{x \rightarrow 0} x \ln x = 0$.
\item Determine the limit of $f(x)$ as $x$ tends to $+\infty$.
\item Prove that, for all strictly positive real $x$, $f'(x) = u(x)$, where $u(x) = \frac{x^2 - 5x + 4}{x^2}$.
\end{enumerate}
Deduce the table of variations of the function $f$ by specifying the limits and particular values.