When $\lim_{x \to\infty} f(x)$ exists and $\lim_{x \to\infty} f'(x) = 0$, can $\lim_{x \to\infty} f''(x) \neq 0$? Let $f$ be a twice differentiable function such that $\lim_{x \rightarrow \infty} f(x)$ is finite and $\lim_{x \rightarrow \infty} f'(x) =
0$. Is it possible that $\lim_{x \rightarrow \infty} f''(x) \neq 0$?

It seems to work to take $f'(x)=\sin(x^2)/x$.
Then it has limit $0$, and $f''(x)=\frac{2x^2\cos(x^2)-\sin(x^2)}{x^2}$, which has no limit in infinity.
As for $f(x)$, according to Wolfram Alpha it is $Si(x^2)/2$ where $Si$ is the sinus integral, so it has a finite limit which is $\pi/4$.
