Defining $n$ for $\lim_{x\to \pm \infty}\frac{1}{x^{n}}=0$.
My notes say that $n$ must be a positive integer, yet there are questions that involve radicals, or answers to some limit questions that make me believe $\lim_{x\to \pm \infty}\frac{1}{x^{\frac{1}{r}}}=0$ where $r$ is a positive integer. Is this limit true though? Can $n$ be instead a positive real number such as $n=\frac{1}{r}$?
You should make sure that the expression makes sense for your choice of $n$. Especially $\lim_{x\to -\infty}1/x^n$ is a bit troublesome if $n$ isn't an integer. I think that's what your notes are referring to.
(You can make sense of $x^n$ for $n\notin \Bbb Z$ and $x<0$, but it takes some work using complex numbers, and there is no canonical way of doing it.)
Once we have limited ourselves to $n\in \Bbb Z$, the answer $n = 1$ is pretty immediate.