Suppose that $\lim_{x\to \infty} f'(x) = a$. Is it true that $\lim_{x\to \infty} {f(x)\over x} = a$

In case L'Hopital's Rule is not allowed :

Let $\epsilon > 0$, Then there exist larg enough $M >0 $ such that $ a-\epsilon \leq f'(x) \leq a+ \epsilon $, for all $ M\leq x $. Now Apply mean value theorem for $f$ on $[M , x ]$ then there exist $y_x \in [M, x]$ such that $$ a- \epsilon \leq f'(y_x) = \frac{f(x) - f(M)}{ x - M} \leq a+ \epsilon $$

Now letting $x \to + \infty$ we get $$ a - \epsilon \leq \limsup_{x \to + \infty} \frac{f(x)}{ x} \leq a+ \epsilon $$ for all $\epsilon > 0$ which implies $ \limsup_{x \to + \infty} \frac{f(x)}{ x} =a$, similarly $ \liminf_{x \to + \infty} \frac{f(x)}{ x} = a$