Analysis question limit $\lim_{n \to \infty} (-1)^n$

Question

Can someone please explain to me how we should evaluate this limit, I just know that it's indeterminate form and I tried to use L'Hospital's rule but I couldn't do it here what I have done $$\lim_{n \to \infty} (-1)^n$$ $$=e^{n\ln(-1)}$$ But $\ln$ of $-1$ doesn't exist because the domain issue of $\ln$.

Answer 1

Since$$(-1)^n=\begin{cases}-1&\text{ if $n$ is odd}\\1&\text{ if $n$ is even,}\end{cases}$$your sequence has a subsequence which converges to $1$ and it has a subsequence that converges to $-1$. Therefore, it diverges.

Answer 2

Note that, the formula $ x= e^{\ln x}$,which you applied only applies to $x>0$. You can directly check odd and even subsequences to see where they converge. If both converge to same limit, the sequence is Convergent, otherwise not.