For what real values of $p$ does $\lim_{x\rightarrow\infty}\frac{1}{x^p+x^{-p}}$ exist and is finite?

Question

For what real values of $p$ does exist and is finite the limit $\lim_{x\rightarrow\infty}\frac{1}{x^p+x^{-p}}$?

I know that $\frac{1}{x^p+x^{-p}}\leq\frac{1}{2}$. I think that this happens only for $p=0$ and I have a feeling this is the only value, but I am unsure on how to develop my response solely from this information. Any help?

Answer 1

The limit is $0$ for $p \neq 0$ and $\frac 1 2$ for $p=0$. [For example, if $p>0$ then $x^{p} \to \infty$ and $x^{-p} \to 0$ so $x^{p}+x^{-p} \to \infty$, etc].