Chapter 12, Problem 4CT

To determine

To find:The approximate limit up to four decimal places of function $\underset{x\to 0}{\lim }\frac{\sin 3x}{x}$ using graphing utility and verify the answer numerically.

Answer to Problem 4CT

Theapproximate limit up to four decimal places of function $\underset{x\to 0}{\lim }\frac{\sin 3x}{x}$ is 2.9997.

Explanation of Solution

Given information:The given function is $\underset{x\to 0}{\lim }\frac{\sin 3x}{x}$ .

Calculation:

To approximate the limit graphically, graph the function $f\left(x\right)=\frac{\sin 3x}{x}$ and follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $\frac{\sin 3x}{x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{sin}}\boxed{3}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\sin \left(\text{3X}\right)/\text{X}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown in Figure (1).

  

Figure (1)

Select two points on the graph as $\left(0.008,2.9997\right)$ and $\left(-0.008,2.9997\right)$ . The approximate limit as the average of the y -coordinates of above two points is:

  $\underset{x\to 0}{\lim }\frac{\sin 3x}{x}=2.9997$

The approximate limit is 2.9997.

To verify the result, create a table of values as shown below:

$x$	$-0.001$	$-0.008$	0	0.001	0.008
$f\left(x\right)$	3	2.9997	undefined	3	2.9997

Therefore, approximate limit up to four decimal places of function $\underset{x\to 0}{\lim }\frac{\sin 3x}{x}$ is 2.9997.