Chapter 11.1, Problem 21E

To determine

To create: a table for the given function and find the limit and graph the function.

Answer to Problem 21E

X	$f\left(x\right)$
-0.1	-0.09966711
-0.01	-0.009999667
-0.001	-0.0009999997
0	Indeterminate
0.001	0.0009999997
0.01	0.009999667
0.1	0.09966711

  $\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{x}=0$

Explanation of Solution

Given:

  $\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{x}$

Calculation:

Let $f\left(x\right)=\frac{{\sin }^{2}x}{x}$

Use graphing calculator to create a table of values of $f\left(x\right)$ for x around 0, and see if the function appears to approach a particular value as x approaches 0 from both the left and right.

X	$f\left(x\right)$
-0.1	-0.09966711
-0.01	-0.009999667
-0.001	-0.0009999997
0	Indeterminate
0.001	0.0009999997
0.01	0.009999667
0.1	0.09966711

Here, $f\left(x\right)$ takes values close to 0 as x takes values close to 0, even though $f\left(x\right)$ is not defined there. Thus, $\underset{x\to 0}{\lim }f\left(x\right)=0$ . This can also be seen from the graph:

  

Conclusion:

Therefore, the value of the given function is $\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{x}=0$

The table is created for the given function.