Suppose that $\lim_{x\to\infty} f '(x) = 0$. Using Mean Value Theorem, show that the limit $\lim_{x\to\infty} f (x) / x = 0$ exists and equals $0$. [duplicate]

Been stuck on this problem for a few days now. It is also given that $f'(x)$ exists for all $x$, and $f'(x)$ can be finite or infinite. My best attempt now is the following, although I'm not 100% convinced:

By MVT, $\exists c_x\in (a,x)$, where $a\in\mathbb{R}$ and $f(a)\in\mathbb{R}$ is finite, s.t. $f'(c_x) = \frac{f(x) - f(a)}{x-a}$.

Now, $\lim_{x\to\infty}f'(c_x) = 0$, since we're given that $\lim_{x\to\infty}f'(x) = 0$ (this is the part I'm uncertain about. Can we make this claim?)

Then $0 = \lim_{x\to\infty}f'(c_x) = \lim_{x\to\infty}\frac{f(x) - f(a)}{x-a} = \lim_{x\to\infty}\frac{f(x)}{x-a} - \frac{f(a)}{x-a} = \lim_{x\to\infty}\frac{f(x)}{x-a} = \lim_{x\to\infty}\frac{f(x)}{x}$

So $\lim_{x\to\infty}\frac{f(x)}{x} = 0$.

Does this proof work? Can we just state that $\lim_{x\to\infty}f'(c_x) = 0$?