Prove: if $\displaystyle f(x)$ is differentiable on $\displaystyle (a,\infty)$ and $\displaystyle \lim_{x\to \infty} f'(x)=0$, then $\displaystyle \lim_{x\to \infty} \frac {1}{x}[f(2x)-f(x)]=0$.

How do I approach such an exercise? Any ideas would be appreciated.