Chapter 11.4, Problem 50ES

To determine

(a)

To prove: Use $\text{L'Hopital's }$ rule to prove that for all integers n = 1, $\underset{x\to \infty }{\text{lim}}\frac{{\log }_{b}x}{x^{1/n}}=0.$

To determine

(b)

Use the result of part (a) and the definitions of limit andof O -notation to prove that ${\log }_{b}x$ is

$O\left(x^{1/n}\right)$ for anyinteger n = 1.