Chapter 2.2, Problem 62E

To determine

Tochoose:The correct option for the value of $\underset{x\to 0}{\lim }\left(\frac{\cos \left(2x\right)}{x}\right)$ .

Answer to Problem 62E

The correct choice is (E).

Explanation of Solution

Given information:

The limit is $\underset{x\to 0}{\lim }\left(\frac{\cos \left(2x\right)}{x}\right)$ .

To graph a function $y=\frac{\cos \left(2x\right)}{x}$ , 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 $y=\frac{\cos \left(2x\right)}{x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{COS}}\boxed{2}\boxed{\text{X}}\boxed{)}\boxed{)}\boxed{÷}\boxed{\text{X}}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\cos \left(2\text{X}\right)\right)/\text{X}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from graph of function $y=\frac{\cos \left(2x\right)}{x}$ tends to $-\infty$ as $x$ to $0$ from the left side and $y=\frac{\cos \left(2x\right)}{x}$ tends to $\infty$ as $x$ to $0$ from the right side.

So, $\underset{x\to 0^{-}}{\lim }f\left(x\right)\ne \underset{x\to 0^{+}}{\lim }f\left(x\right)$ . The value of $\underset{x\to 0}{\lim }\left(\frac{\cos \left(2x\right)}{x}\right)$ does not exist.

Therefore, the correct choice is (E).