# Limit of a power function with imaginary power

Is there is any precise notion of the limit

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

where $\mu$ is purely imaginary?

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Of course this is to complete things, since the limit is $0$ if real part of $\mu$ is positive, and limit is complex infinity if real part of $\mu$ is negative. –  GEdgar Aug 16 '11 at 16:28
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## 1 Answer

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.

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