Limit points of $z^n$ ($z\in\mathbb{C}$)?

Question

The exercise is to find the limit points of $z^n$ where $z\in\mathbb{C}$ is a complex number?

However, if $z=-1$ we have the limit points $1$ and $-1$, for $z=1$ we have the limit point $1$ and for $z=2$ we have no limit point. So how can you determine the limit points of $z^n$ in general?

Answer 1

Hint: You can determine the limit points easily if $|z|\neq1$ (you have two cases to consider). Now consider what happens when $|z|=1$, which is equivalent to $z=e^{\pi i\theta}$. The cases depend on the rationality of $\theta$, so how does it affect the limit points?