Given the function $f(x)=\lim_{n\to\infty} x^n$, what its domain is?
I have searched plenty materials implicitly contain this problem. They give out 2 different answer: $x\in [0,+\infty)$ and $x \in (-\infty, +\infty)$.
I have also learned conditions about $n=p/q$. Seemingly this doesn't help my problem. As $\infty$ is neither rational nor irrational.
I want to learn reasons of the answers, and citations.
In fact this question is extracted from $f(x) = \lim_{n\to\infty} \frac { x^{2n-1} + x} {x^{2n}+1}$. My answer book says the domain is $x\in\mathbb{R}$.
In my brain I consider $f$'s domain is subject to $x^{2n}$, so come up with this question.
Moreover, I want to learn the generalized case. If a function is like $\lim_{n\to\infty} g(x^n)$, how I should consider the $x^n$ part.