IMC 1995 · Problem 3

Let $f$ be twice continuously differentiable on $(0, +\infty)$ such that $\lim\limits_{x \to 0+} f'(x) = -\infty$ and $\lim\limits_{x \to 0+} f''(x) = +\infty$. Show that

$$\lim_{x \to 0+} \frac{f(x)}{f'(x)} = 0.$$