Chapter 9.3, Problem 13E

To determine

Toshow: that $\ln x,\log \sqrt{x}$ grows at the same rate.

Explanation of Solution

Given information:

The functions are $\ln x,\log \sqrt{x}$ .

proof:if the ration of the both function at limit of $x\to \infty$ exist, then $\ln x,\log \sqrt{x}$ grows at the same rate.

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{\log \sqrt{x}}{\ln x}=\underset{x\to \infty }{\lim }\frac{\log x}{2\ln x}\text{ [}\log a^x=x\log a\text{]}\\ \underset{x\to \infty }{\lim }\frac{\log \sqrt{x}}{\ln x}=\underset{x\to \infty }{\lim }\frac{\ln x}{2\ln x\left(\ln 10\right)}\text{ [}\log x=\frac{\ln x}{\ln 10}\text{]}\\ \underset{x\to \infty }{\lim }\frac{x^2+\sin x}{x^2}=\frac{1}{2\ln 10}\end{array}$

Thus, $\ln x,\log \sqrt{x}$ grows at the same rate.