Compare growth of function and its derivative.

Question

Suppose $f:\mathbb R \rightarrow \mathbb R$, $f(x)<O(x)$ (i.e. has slower than linear growth), $\lim_{x\rightarrow \infty} f'(x)=0$. Is it possible to show that there exists a $\delta$ such that for $|x|>\delta$, $x f'(x) < f(x)$ for all functions $f$?

Answer 1

This is not true. For example, take $f(x)=\frac{1}{x}\sin x$. Then $f'(x)=\frac{1}{x}\cos x - \frac{1}{x^2}\sin x \to 0$ as $x\to \infty$. However, $xf'(x) = \cos x - \frac{1}{x} \sin x \not\to 0$ while $f(x) \to 0$.