Prove or disprove: If for $f:(a,b) \to \mathbb R$ the limit $\lim_{x \to a^+} \frac{f(x)-f(a)}{x-a}$ is equal to $- \infty$, then there is $a
It seems it should be true, since I have trouble picturing a counterexample. On the other hand, we cannot use MVT or something like that here, so I cannot prove it either.
Any ideas?