Exercise 6, § 4

Let $f$ be a convex function on an interval $]a,\ +\infty[$; show that $f(x)/x$ has a limit (finite or equal to $+\infty$) as $x$ tends to $+\infty$; this limit is also that of $f'_d(x)$ and of $f'_g(x)$; it is $> 0$ if $f(x)$ tends to $+\infty$ as $x$ tends to $+\infty$.