Remark

A similar argument, based on th. 1, shows that if $f$ is a real function such that $f'_d(x)$ tends to $+\infty$ at the point $x_0$ then the ratio

also tends to $+\infty$, and conversely; if moreover $f$ has a finite limit at the point $x_0$ (which is not a consequence of the present hypothesis), then the function $f$ extended by continuity to $x_0$ has a derivative equal to $+\infty$ at this point.