# Improper integral for $\ f \rightarrow \infty$

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?

• the function must go to zero as $x$ goes to infinity if the improper integral is going to have any chance of existing. – Thoth Nov 15 '15 at 23:31
• Thanks a lot for the answer! – Gianolepo Nov 15 '15 at 23:40

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.