The Limit Comparison Test for Improper Integrals

Question

The test says that if $0\le f(x)\le g(x)$ and if $L=\lim_{x\to \infty}\frac{f(x)}{g(x)}$ When $0<L<\infty$ so if $g(x)$ converges so does $f(x)$ and if $g(x)$ diverges so does $f(x)$.

What can be said if $L=0$ or $L$ approaches $\infty$?

Answer 1

$L=0$ means $g$ grows faster than $f$. Then the only valuable information you can draw is about $g$: $f$ converges/diverges $\Rightarrow g$ converges/diverges.

$L=\infty$ means instead that $f$ grows faster than $g$, but this cannot happen because of the first condition $f(x) \leq g(x)$ unless they are both zero and $L$ is indeterminate.