calculus - $f$ is an unbounded monotonic increasing function: $lim limits_{x to infty} frac{1}{x}int_{0}^{x}f(t)dt=infty$

This is not homework.

I'd love your help proving that if $f$ is an unbounded monotonic increasing function, then $$\lim_{x \to \infty} \frac{1}{x}\int_{0}^{x} \ f(t)\, dt=\infty.$$ I want to use it in a couple of proofs, but I can't prove it by myself.

Thanks a lot.

Answer

I think the l'Hôpital's rule may be helpful here if your $f$ satisfies the requirements of the rule. And say the antiderviative of $f$ is $F(x)$, then

$$ \lim_{x\to\infty}\frac{1}{x}\int_{0}^{x}f(t)dt = \lim_{x\to\infty}\frac{F(x) - F(0)}{x} = \lim_{x\to\infty}f(x) .$$

Since your tag is calculus, not something like real analysis, I assume your function is a relatively normal function involving Riemann integral.