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

Question

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: $$z=x+jy\,\,\,x,y\in\mathbb{R}$$ $$\lim_{(x,y)\to(\infty,\infty)}e^{-(x+jy)}=0$$

Answer 1

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.

Answer 2

Of course not ! take $x_n=2in\pi$ and $y_n=n$ to conclude the non existence.

Answer 3

Try putting in $z_n=-n$ which satisfies $|z|\rightarrow\infty$. So we're having a bad day.

Answer 4

Approach along the negative $x-$axis starting from the origin i.e. $x\rightarrow-\infty$ and $y=0$.