Showing that if $\lim_{x\to\infty}f'(x)=L$ then $\lim_{x\to\infty}\frac{f(x)}{x} = L$.

Hint: If for $x > N > 0$, $ f'(x) < L+\epsilon$, then for such $x$, $f(x) < f(N) + (x-N) (L + \epsilon)$ and $$\dfrac{f(x)}{x} < L + \epsilon + \dfrac{f(N)- N(L+\epsilon)}{x} $$ What's the limit of the right side as $x \to \infty$?

Similarly in the other direction...