Limit at Infinity of Polynomial over Complex Exponential

Theorem
Let $n \in \N$.

Let $P_n \left({x}\right)$ be a real polynomial, of degree $n$.

Let $e^z$ be the complex exponential, where $z = x + i y$.

Then:


 * $\displaystyle \lim_{x \mathop \to +\infty} \frac {P_n \left({x}\right)}{e^z} = 0$

Proof
Let $\epsilon > 0$.

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


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

But:

This means it is sufficient to find an $M > 0$ such that:

The result then follows from Exponential Dominates Polynomial.