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Proposition: Limits of Polynomials at Infinity

Let $p:\mathbb R\to\mathbb R$ be a real polynomial with the degree $n$, i.e. $$p(x):=a_nx^n + \ldots + a_1x + a_0.$$

Then there are the following limits of $p$ at infinity:

$$\begin{array}{rcl}\lim_{x\to+\infty}p(x)&=&+\infty\\\lim_{x\to-\infty}p(x)&=&\cases{+\infty,& if$n$is even\\-\infty,&if$n$is odd.}\end{array}$$