TITLE: If $ f\geq0 $ and $ \intop_{0}^{\infty}f\left(x\right) $ converge, and $ \intop_{0}^{\infty}f'\left(x\right) $ converge, does it mean that:
QUESTION [7 upvotes]: Let $ f $ be non negative function, and differntiable such that $ f' $ is continious and $ \intop_{0}^{\infty}f\left(x\right),\intop_{0}^{\infty}f'\left(x\right) $ both converge.
Is it true that $ \lim_{x\to\infty}f'\left(x\right)=0 $?
I know that $ \lim_{x\to\infty}f\left(x\right)=0 $ for sure in this conditions. But Im not sure how to tell something about the limit of the deriviative.
Thanks in advance

REPLY [4 votes]: For $x \geqslant 1$, take $\displaystyle f(x) = \frac{2+\sin x^3}{x^2}$ and extend as a C1 function on $[0,1]$.
Note that $f'(x) = 3 \cos x^3 - (4+2 \sin x^3)/x^3$ and $\lim_{x \to \infty} f'(x)$ does not exist.