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I'm attempting to classify the singularities of $f(z) = z^3e^\frac {1}{z}$ using limits, as I haven't learned how to find Laurent expansions yet. To show that the function has an essential singularity at $z=0$ I need to find a sequence that makes the limit finite and one that makes it approach infinity. I found one to make it approach infinity, but I'm struggling to come up with one to make it finite. Any hints are appreciated!

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  • $\begingroup$ A sequence....converging to what ? $\endgroup$ – DonAntonio Apr 8 '17 at 17:37
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What about

$$z_n:=\frac in\implies |f(z_n)|=\left|-\frac i{n^3}e^{-ni}\right|=\frac1{n^3}\xrightarrow[n\to\infty]{}0\;\;\;?$$

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