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?

The integral 
$$\int_{1}^\infty f(x)dx$$
is defined as
$$\lim_{b\to\infty} \int_1^b f(x)dx$$
so it should be simple for you to show that if there exist $x_0\geq 1$ and $M>0$ such that $f(x)>M$ for all $x>x_0$ (this is of course true of $\lim_{x\to\infty} f(x)=\infty$), then $$\int_{1}^\infty f(x)dx = \infty.$$
