If the limit of a function exists as $x$ tends do infinity, must its derivative also exist?

Question

I thought of a scenario where: if $ \lim_{x-> \infty}f(x)$ exists, which I can only imagine is a constant or asymptotic function, then it also seems like the derivative f' must also exist, but I don't know how to prove this either way. I guess if there is just a counterexample that would be easier than proving it. Is there an asymptotic function that somehow has a divergent derivative?

Answer 1

No, take for example $f(x)=\frac{\sin{x^2}}{x}$. Then $\lim_{x\to\infty} f(x) = 0$, but $f'(x) = 2 \cos{x^2} - \frac{\sin{x^2}}{x^2}$ and so $\lim_{x\to\infty} f'(x)$ does not exist.