SCHWEITZER 1968 Problem 9

Let $f(x)$ be a real function such that $$\lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1$$ and $|f''(x)|\leq c|f'(x)|$ for all sufficiently large $x$ . Prove that $$\lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.$$

P. Erdos