# Odd degree polynomial limit.

I am asked to prove that if $p(x)\in \mathbb R[x]$ with the condition that $\deg (p(x))$ is odd then $\lim_{x\to \infty} p(x) = -\lim_{x\to-\infty} p(x)$

My approach: I want to show that either $$\lim\limits_{x \to \infty} p(x)=\infty$$ or$$\lim\limits_{x \to \infty} p(x)=-\infty$$ then $$\lim\limits_{x \to -\infty} p(x)=-\infty$$ or $$\lim\limits_{x \to -\infty} p(x)=\infty$$ respectively. Is this correct, is this idea worth considering?

Thank you for help beforehand.

• I'm putting a minus sign in the statement to make it true. :-) – Sammy Black Mar 19 '13 at 22:27
• For proof, you may want to split into two cases, (i) lead coefficient positive; (ii) negative. (Once you have done the first, the second is trivial.) – André Nicolas Mar 19 '13 at 22:30

Hint: Prove $$\lim\limits_{x\to\infty}\frac{p(-x)}{p(x)}=-1$$