Suppose that $\lim_{x\to \infty} f'(x) = a$. Is it true that $\lim_{x\to \infty} {f(x)\over x} = a$ Suppose that $\lim_{x\to \infty} f'(x) = a$. Is it true that $\lim_{x\to \infty} {f(x)\over x} = a$
If so, can you prove it? Thanks!

HINT
Let assume $a \gt 0$. Then, because $\lim_{x\to \infty} f'(x) = a \gt 0$ there is $M \gt 0$ such that $f'(x) \gt 0$ for all $x \ge M$. Therefore $f$ is strictly increasing on $[M, +\infty)$ and unbounded (otherwise $\lim_{x\to \infty} f'(x) =0$). It follows that $\lim_{x\to \infty} f(x) = +\infty$ and we can apply L'Hospital to $\lim_{x\to \infty} {f(x)\over x}$
Now, let's assume $a=0$. Let $g(x) = f(x) + x$. Then $\lim_{x\to \infty} g'(x)=1$ therefore $\lim_{x\to \infty} {g(x)\over x}=1$ and from here $\lim_{x\to \infty} {f(x)\over x}=0$
