Textbook Problem

Exercises 49 and 50 use L’Hôpital’s rule from calculus.

50. a. Let b be any real number greater than 1. Use L’Hôpital’s rule to prove that for every integer $n\ge 1$ , $\underset{x\to \infty }{\lim }\frac{{\log }_{b}x}{x^{1/n}}=0.$

b. Use the result of part (a) and the definitions of limit and of O-notation to prove that ${\log }_{b}x$ is $O\left(x^{1/n}\right)$for any integer $n\ge 1$ .