# Intuition on Lim inf and lim sup

I am looking at trying to justify an example to myself (I'm not trying to prove anything I just want to have an idea of what is going on). If I consider $$r\in\mathbb{R}, r\neq 0$$ what can we say about the following:

$$\lim\inf\limits_{n\to\infty} (-1)^n(r^n-r^{-n})$$

$$\lim\sup\limits_{n\to\infty} (-1)^n(r^n-r^{-n})$$

$$\lim\limits_{n\to\infty} (-1)^n(r^n-r^{-n})$$

So for obvious cases, like $$r=1$$ this is a constant sequence so it converges and theyre all the same. For example, if $$0 it seems this would diverge, but can we say anything for the lim inf and lim sup? I'm just trying to get a handle on cases where there is no limit how to justify intuitively what to look for.