Q:

$\lim_{|z|\to\infty}e^{-z}$

I was wondering if the following limit can be evaluated:
$$\lim_{|z|\to\infty}e^{-z}$$
My first instinct was not, as $e^{-z}\to 0$ is only true for $\Re(z)\to\infty$ since $\Im(z)$ would just fluctuate with $0\le e^{\Im(z)}\le1$. So my question is to define this limit correctly and assume it tends to zero would we have to say something like:
$$z=x+jy\,\,\,x,y\in\mathbb{R}$$
$$\lim_{(x,y)\to(\infty,\infty)}e^{-(x+jy)}=0$$

A:

No, and very badly so. The exponential function has an essential singularity at infinity, which means it assumes all but finitely many values of $\mathbb C$ in every neighborhood (so infinitely often) of the singularity.

