Improper integral for $\ f \rightarrow \infty$

Question

Maybe it's a stupid question but i just want to be sure of that. If a function approaches $\infty$ as $\mathrm x \rightarrow \infty$ it is useless trying to evaluate its improper integral from (for example) 0 to $\infty$ as it will surely diverge, is it correct?

Answer 1

This is an informal, but (in my opinion) intuitive way of understanding this question:

If we consider the improper integral as a continuous summation of the area under the curve, we realize that the only way this area could be finite is if we add less and less area as we go.

If you are considering here $$ \int_0^{\infty}f(x)dx:f(x):\mathbb{R}\to\mathbb{R}$$ where $$\lim_{x\to\infty}f(x)\to k\ne0$$

then for $x$ sufficiently large, we essentially have an infinitely extending rectangle of height $k$ fitting under the curve. This could never converge, because the area is definitely infinite.

Of course, if the curve gets "higher" as we take $x$ towards infinity, then we can fit an arbitrarily large rectangle under the curve. So the integral, once again, can not converge to a finite value.