# Proof limit even degree polynomial

Hello I'm trying to prove the following fact

Let $$f(x) = x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ with $$n \in \mathbb{N}$$ and $$a_{n-1},\ldots,a_0 \in \mathbb{R}$$. If $$n$$ is even, then we have $$\displaystyle\lim_{x\to+\infty}f(x)=+\infty$$ and $$\displaystyle\lim_{x\to -\infty}f(x)=+\infty$$.

This seems quite intuitive since the highest degree of the polynomial and any number squared is even. However this is obviously not a proof. So my second thought was that there could be a way to prove that $$f(x)$$ is not injective, meaning that some values could have more than one corresponding $$x$$ value. However, this also does not quite hold (I think) for any polynomial. However this is where I'm stuck. How can I properly prove it ?

Hint: Use the fact that $$f(x)=x^{n}\left[1+\frac {a_{n-1}} x+\frac {a_{n-2}} {x^{2}}+\cdots+\frac {a_0} {x^{n}}\right].$$
• Oh so i can just say that the limit of each of these terms is 0 (except for the 1 of course) and then go on from there saying $x^n$ with $n$ even. Should I try proving that using the property that $f(x) = f(-x)$ for all $x$? Although I'm not quite sure how that would be done Nov 9, 2020 at 9:31
• If $g(x) \to \infty$ and $h(x) \to 1$ then $g(x)h(x) \to \infty$. Nov 9, 2020 at 9:37