Complex Exponential Tends to Zero

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

Then:


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

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

Proof
By the definition of limit, this implies:


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

where $e^M$ is the real exponential of the real variable $M$.

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The result follows from real exponential tends to zero.