Intuitively, I understand that if I take limits to infinity, then grows really large and grows very small so the difference is less than zero. If I take the limits to negative infinity then the opposite happens. Using the intermediate value theorem, there must be an intersection, and since they are strictly increasing/decreasing, only one intersection happens.
My question is how do I apply the intermediate value theorem here? I don't have the interval to apply it on. I don't know when crosses and therefore can't take any interval. Or is there some sort of axiom applied here that I am missing?