When I type in $\lim\limits_{x\to0}x^{\frac1x}$ in Wolfram Alpha, it gives $e^{2i 0\text{ to }\pi}\infty$ for the one sided left limit. Can anybody explain what it means?
Wolfram Alpha $e^{2i 0\text{ to }\pi}\infty$
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0It seems a set of different infinities in the complex plane. The $e^{i\alpha}$ for $\alpha\in[0,2\pi]$ is the set of directions, and $\infty$ is the modulus. – 2017-01-18
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0Ah, so this means that if we express $x^\frac{1}{x} = re^{i\varphi}$ and let $x\to 0$ then $r\to\infty$, moreover, for each angle $\phi$ we can achieve $\varphi\to\phi$ if $x\to 0$ along a suitable sequence? All this for some branch of the logarithm? @Masacroso – 2017-01-18
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0I suppose, Im not sure, but it seems that is something like this. – 2017-01-18