Proving $\lim_{x\to\infty}\frac{f(2x)-f(x)}{f'(x)}=\infty$, when $f'(x)$ is positive and monotonically increasing

Question

I have an assignment that I need a hint or help with.

Given a $f(x)$ is differentiable. Assuming $f'(x)$ is a positive monotonically increasing function, prove that: $$\lim_{x\to\infty}\frac{f(2x)-f(x)}{f'(x)}=\infty$$

I have tried using the definition of limit to solve it, but can't figure out how to proceed from there.

Because the function is positive, monotonical, and increases I know that $f(x)$ is increasing as well, and there for $f(2x)\geq f(x)$ for every $x$. Now I just need to prove that the upper value is always greater than the $f'(x)$.

How should I approach this?

Thank you.

Answer 1

$f(2x)-f(x) =f'(t)(2x-x)\geq f'(x)(2x-x)$ for some $t$ between $x$ and $2x$ by MVT. Hence $\frac {f(2x)-f(x)} {f'(x)} \geq \frac {xf'(x)} {f'(x)}=x \to \infty$.