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Inspired from this question I want to ask if the converse also holds.

Let $f:(a, \infty) \to\mathbb {R} $ be differentiable on $(a, \infty) $.

If for every $h>0$ the limit $\lim\limits_{x\to\infty} \dfrac{f(x+h) - f(x)} {h} $ exists and is equal to $L$ does the limit $\lim\limits_{x\to\infty} f'(x) $ exist?

Using the mean value theorem it does not look like there is too much room for $f'(x) $ to oscillate, but I am not able to formulate the argument properly (at least for now) and therefore I guess that it might be possible that the limit of $f'(x) $ does not exist. But I have not found a counter-example.

The converse for the second part in linked question does not hold. That is, if $\lim\limits_{x\to\infty} \dfrac{f(x)} {x} $ exists it does not necessarily imply that $\lim\limits_{x\to\infty} f'(x) $ exists (check with $f(x) =\sin x$).

Paramanand Singh
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The function $f(x)=\frac{\sin(x^2)}{x}$ on $[1,+\infty[$ is a counter-example. As $f(x)\to 0$ if $x\to +\infty$, the limit exists for all $h>0$ (and is $0$). But the derivative of $f$ is $f^{\prime}(x)=-\frac{\sin(x^2)}{x^2}+2\cos(x^2)$, and has no limit if $x\to +\infty$.

Kelenner
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