# Essential singularity

## Homework Statement

http://en.wikipedia.org/wiki/Essential_singularity

What is the best way to prove that e^{1/z} has an essential singularity at z=0? I have tried showing that
$$\lim_{z\to 0} z^k e^{1/z}$$
does not exist for any natural number k, but I couldn't get it.

## The Attempt at a Solution

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Dick
Homework Helper
Why couldn't you get it? The limit doesn't even exist as you approach 0 along the positive real axis.

Why couldn't you get it? The limit doesn't even exist as you approach 0 along the positive real axis.
How do you prove that? I tried using the definition of e^x

$$\lim_{x\to 0} \lim_{k\to \infty}\sum _{n=0}^k \frac{x^{-n+k}}{k!}$$

But the double limit makes that especially hard to evaluate.

Dick