Does $f, f' \in L^1([0, \infty))$ imply that $\lim_{x \to \infty} xf(x) = 0$? Does $\int_0^\infty |f(x)| \, dx$ and $\int_0^\infty |f'(x)| \, dx$ being finite imply that $\lim_{x \to \infty} xf(x) = 0$? 
(Context: I am working through an analytic number theory textbook. In a lemma, the book assumes the conditions on $f$ and $f'$. The term $xf(x)$ appears from an integration by parts, but is tossed out, which must mean that $\lim_{x\to\infty} xf(x) = 0$. However, I've tried to prove this and was not successful. I also cannot think of any counterexamples.)

One way to think about problems of such type is in terms of linear interpolation.
Define your function $f(n)=a_n>0$ let $f$ to be linear between $[n,n+1].$ Then, $f\in L_1$ is equivalent to $\sum_{n=1}a_n<\infty$ whereas  $f'\in L_1$ is equivalent to $\sum_{n=1}|a_{n+1}-a_n|<\infty.$ Now it is clear that the standard example of series mentioned by Nick satisfies both conditions. The only thing left is to "smoothen" $f$ at the corners to make sure that $f'$ exists.
