Chapter 11.2, Problem 43E

To determine

To graph: the given function and find the limit.

Answer to Problem 43E

The value of the limit is $\underset{x\to 0}{\lim }\frac{\sin 2x}{x}=2$

  

Explanation of Solution

Given:

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

Calculation:

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

Estimate the limit:

  

use a graphing utility to graph the function. Using the zoom and trace features choose two points on the graph such as:

  $\begin{array}{l}\left(-0.00001,2.00000\right)\\ \left(0.00001,2.00000\right)\end{array}$

  $\underset{x\to 0}{\lim }\frac{\sin 2x}{x}=\frac{2.00000+2.00000}{2}=\frac{4}{2}=2$

Because the $x$ -coordinates of the two points are equidistant from $0$ , approximate the limit to be the average of the $y$ -coordinates.

Conclusion:

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