If $\lim_{x\rightarrow \infty}f(x) = \lim_{x\rightarrow \infty}f''(x) = 0$, then $\lim_{x\rightarrow \infty}f'(x)=0$

Question

The problem is stated as the following:

Given that $f(x)$ is twice differentiable in $\mathbb{R}$, show that as $\lim_{x\rightarrow \infty}f(x) = \lim_{x\rightarrow \infty}f''(x) = 0$, then $\lim_{x\rightarrow \infty}f'(x)=0$

I have tried to solve the problem using Lagrange's mean value theorem, but it doesn't seem to simplify the problem. Then I thought that I might be able to use the squeeze theorem somehow, but I don't really know how to get the first derivative in an inequality.

If you have any tips that can help me solve the problem, I'd be glad if you could share them. Thanks!