Limit of a power function with imaginary power

Question

Is there is any precise notion of the limit

$\lim_{x\rightarrow0^+}x^\mu$

where $\mu$ is purely imaginary?

Answer 1

Let $x=e^{-n\pi}$ where $n$ is a (large) positive integer. Let $\mu=-i$.

Then $x^\mu=e^{in \pi}$. By Euler's Formula, if $n$ is odd, then $e^{in \pi}= -1$, and if $n$ is even then $e^{in \pi}=1$. As $n \to \infty$, $x \to 0^+$, so the limit does not exist.

Other choices of positive $x$ arbitrarily near $0$ can give us any complex number on the unit circle.