Complex Exponential Tends to Zero

Theorem
Let $\exp z$ be the complex exponential.

Then:


 * $\displaystyle \lim_{\operatorname{Re}z \to -\infty} \exp z = 0$

where $\operatorname{Re}z$ is the real part of $z$.

Proof
Let $z = x + iy$.

Then:

By the definition of limit, this implies:


 * $\displaystyle \lim_{x \to -\infty} e^x = \lim_{\operatorname{Re}z \to -\infty} \exp z$

where $e^x$ is the real exponential.

The result follows from real exponential tends to zero.