Chapter 13.4, Problem 18E

To determine

To find : The limit $\underset{x\to \infty }{\lim }{\sin }^{2}x$.

Answer to Problem 18E

The limit $\underset{x\to \infty }{\lim }{\sin }^{2}x$ does not exist.

Explanation of Solution

Given information:

The given limit is $\underset{x\to \infty }{\lim }{\sin }^{2}x$

Calculation :

Let $f$ be a function defined on some interval $\left(a,\infty \right)$ . Then

  $\underset{x\to \infty }{\lim }f\left(x\right)=L$

means that the value of $f\left(x\right)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently large.

Draw graph of ${\sin }^{2}x$.

  

From the graph of ${\sin }^{2}x$ , it can be observed that as the value of x increases, ${\sin }^{2}x$ attains every value between 0 and 1, so it does not approach any certain number.

Therefore, $\underset{x\to \infty }{\lim }{\sin }^{2}x$ does not exist.

Hence,

The limit $\underset{x\to \infty }{\lim }{\sin }^{2}x$ does not exist.