Chapter 2.2, Problem 69E

To determine

To find: The value of $\underset{x\to \infty }{\lim }\left(\frac{\ln x^2}{\ln x}\right)$ and also explain.

Answer to Problem 69E

The value of $\underset{x\to \infty }{\lim }\left(\frac{\ln x^2}{\ln x}\right)$ is $2$ .

Explanation of Solution

Given information:

The limit is $\underset{x\to \infty }{\lim }\left(\frac{\ln x^2}{\ln x}\right)$ .

Calculation:

Simplify the given limit.

  $\begin{array}{c}\underset{x\to \infty }{\lim }\left(\frac{\ln x^2}{\ln x}\right)=\underset{x\to \infty }{\lim }\left(\frac{2\ln x}{\ln x}\right)\\ =\underset{x\to \infty }{\lim }\left(2\right)\\ =2\end{array}$

Therefore, the value of $\underset{x\to \infty }{\lim }\left(\frac{\ln x^2}{\ln x}\right)$ is $2$ as the function $\frac{\ln x^2}{\ln x}$ is equal to constant function $f\left(x\right)=2$ .