Rate at which an integral approaches infinity

Question

If $$\lim_{x\to\infty} f(x) = \infty$$ what can be said about the rate at which $$\int_1^\infty f(x) \,dx$$ approaches infinity if $f(x) \geq 1$ for all values of $x$?

Answer 1

All that can be said is that $$\frac{\int_1^x f(y)dy}{x}\rightarrow \infty.$$ No better lower bound can be given, and nothing can be said about the rate at which this goes to infinity since nothing is given about $f$. Indeed, you can construct $f$ so that this ratio goes to infinity as slowly, or as quickly as desired.

Answer 2

Is this all the information given? If so, not much.

$f(x)$ could be $|x| + 1$ or $e^x + 1$. These are two pretty different functions, in terms of the rate at which their integrals grow. You would need more information to say much of anything at all.