Chapter 4 - Section 4.4 - Indeterminate Forms and l''Hospital''s Rule - 4.4 Exercises - Page 312: 74

$\lim\limits_{x \to \infty}\frac{ln~x}{x^p} = 0$

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