Complex Exponential Tends to Zero

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

Then:


 * $\displaystyle \lim_{\operatorname{Re} \left({z}\right) \mathop \to +\infty} e^{-z} = 0$

where $\operatorname{Re} \left({z}\right)$ denotes the real part of $z$.

Proof
Let $z = x + iy$.

Let $\epsilon > 0$.

By the definition of limits at infinity, we need to show that there is some $M > 0$ such that:


 * $\displaystyle x > M \implies \left \vert {e^{-z} - 0} \right\vert < \epsilon$

But:

so we need an $M$ such that:


 * $\displaystyle x > M \implies \left \vert {e^{-x} - 0} \right\vert < \epsilon$

But this is precisely the definition of the limit at infinity of the real exponential.

The result follows from Exponential Tends to Zero and Infinity.