Function and its derivatives at infinity Let $ f: \mathbb{R} \to \mathbb{R}$ be a doubly differentiable function. Suppose that $ lim_{x \to \infty} f(x) = 0 $ and $ lim_{x \to \infty} f''(x) = 0 $. Does this imply that $ lim_{x \to \infty} f'(x) = 0 $?

Barbalat's lemma (proof here): 

If $f$ converges to some $L$ and $f'$ is uniformly continuous over some interval $[a,\infty)$, then $f'$ converges to $0$.

Since $f''$ converges (the value of the limit is irrelevant), $f''$ is bounded over some $[A,\infty)$, thus $f'$ is uniformly continuous over $[A,\infty)$.
