Question about taylor expansion If I was given a function which its derivative is bounded for every $x>0$ (means: $|f'(x)|\le M$), How can I prove that $\lim_{x\to\infty}\frac{f(x)}{x^2}=0$? 

If $\lim_{x\to\infty}f(x)<\infty$, the result holds.
Otherwise, use L'H rule. Since $f'(x)$ is bounded while $x$ is not bounded, you get the result.
