$P(z)$ is a polynomial ($z \in \mathbb{C}$) then $\lim_{|z| \to \infty}|P(z)| = \infty$

Question

If $P(z)$ is a polynomial ($z \in \mathbb{C}$), how does one prove (in the most basic way) that $$\lim_{|z| \to \infty}|P(z)| = \infty?$$

Answer 1

Hint

If $P(z)=a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0\in\mathbb{C}[z]$ and $a_n\ne 0$, then $$\lim_{|z| \to \infty}|P(z)|=\lim_{|z| \to \infty}|a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0|$$ $$=\lim_{|z| \to \infty}|z|^n\left|a_n+\frac{a_{n-1}}{z}+\cdots +\frac{a_{1}}{z^{n-1}}+\frac{a_{0}}{z^n}\right|.$$