Question

Which function dominates x^p or p^x

Answer 1

I'm assuming $p\in\mathbb R$ is fixed. First note that $x^p=p^x$ when $x=p$.

We can rewrite each expression as

$x^p=e^{p\ln x}$
$p^x=e^{x\ln p}$

Now to compare we only need to consider the exponents. It's easy to show that $x>\ln x$ for all $x>0$, so it follows that $x^p$ so long as $x>p$.