Can the limit $$\lim_{x\to 0} x^i = \lim_{x\to 0} \exp( i \log x )$$ be defined? I know that in "classical" sense this equals $$x^i = \cos\log x + i \sin\log x$$ so the limit doesn't exist, but can it be defined in some distribution sense, so it makes some sense if the expression shows up in some integral, or something of the sort?
I apologize if the question is kinda vague; the problem I'm working on includes an integral of a total derivative of $ e^{i f(x) + i \log x}$ from $0$ to $X$, which should exist (based on some other considerations), but I don't know how to evaluate it, since at one boundary I have $\lim_{x\to 0} e^{i f(x) +i \log x}$ ($f(x)$ is complex in general, but at $x\to 0$ it converges to a real value). Any help (however vague) would be greatly appreciated.