Why does $1^{\infty}$ not exist?

Question

In my calculus class, whenever we try to find a limit of a sequence as it approaches infinity and it turns out to be like: $1^{\infty}$. We have to end up using L'Hopital's rule. I don't understand why it has to be L'Hopitaled, can't you just take the limit as the sequence approach $99999$ and the answer would be $1^{99999} = 1$ and you are done?

Why do we have to L'Hopital then?

Answer 1

Consider the following: $(1+1/n)^n \to e,$ while $(1+1/n)^{n^2} \to \infty.$