# Essential singularity

1. Mar 31, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution

2. Mar 31, 2008

### Dick

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

3. Mar 31, 2008

### ehrenfest

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.

4. Mar 31, 2008

### Dick

It's easiest if you take x=1/z for z real. Then the limit becomes lim(x->infinity) e^x/x^k. Now 'everybody knows' e^x approaches infinity faster than any power of x. But if you want to show it, use l'Hopital k times.