Answer
$\lim\limits_{x \to \infty}\frac{ln~x}{x^p} = 0$
Work Step by Step
Suppose that $p \gt 0$
We can use L'Hospital's Rule:
$\lim\limits_{x \to \infty}\frac{ln~x}{x^p}$
$=\lim\limits_{x \to \infty}\frac{(1/x)}{px^{p-1}}$
$=\lim\limits_{x \to \infty}\frac{1}{px^p}$
$= 0$