If the limit of a function when x goes to infinity is infinity, does that mean the integral from 1 to infinity is also infinity? If the limit of a function when x goes to infinity is infinity, does that mean the integral from 1 to infinity is also infinity?
Can we use the above to show that an unsolvable integral is infinity?
edit: Can we also say that if the limit of a function when x goes to infinity is a number then the integral is not infinity?

If $f(x)\to\infty$ as $x\to\infty$, then we have that $f(x)\ge M$ for some $M>0$ when $x>x_0$. Hence, we have that
$$
\int_1^\infty f(x)\mathrm dx
=\int_1^{x_0}f(x)\mathrm dx+\int_{x_0}^\infty f(x)\mathrm dx
\ge\int_1^{x_0}f(x)\mathrm dx+M\int_{x_0}^\infty \mathrm dx=\infty.
$$
