TITLE: Can the limit of a complex number have multiple values?
QUESTION [6 upvotes]: I have been studying "Complex analysis" for a while. In this "limit of complex function" question, I get an answer that's a $4$th root of $-1.$ I change  that form to polar coordinate form and then see that there's more than one value. This is the work that I have done so far:

So,

Can the limit of a complex function has more than one value?
If not, do I need to change an answer that's a fractional exponent to polar form?

REPLY [6 votes]: As has been pointed out, your first equality contains the mistake $$(e^{\pi i})^{1/4}\color{red}=e^{\pi i/4}.$$ The LHS are the four $4$th roots of $-1,$ while the RHS is the single value $\displaystyle\frac{1+i}{\sqrt2}.$

In general, the law $$\large(e^z)^n=e^{nz}$$ holds for $$z\in\mathbb C,\,n\in\mathbb Z$$ and, separately, for $$z,n\in\mathbb R.$$

Multiple values are created not by converting a complex number to polar form per se, but by raising it to the power $\displaystyle\frac1n$ (obtaining its $n$th roots).