Chapter 2.1, Problem 65E

To determine

To find: The value of $\underset{x\to 0}{\lim }\left(x\sin x\right)$ with the help of graph and check the results with the help of sandwich theorem.

Answer to Problem 65E

The value of $\underset{x\to 0}{\lim }\left(x\sin x\right)$ is $0$ .

Explanation of Solution

Given information: The limit is $\underset{x\to 0}{\lim }\left(x\sin x\right)$ .

Calculation:

To find the limit graphically, let $y=x\sin x$ .

To graph a function $y=x\sin x$ , 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 $y=x\sin x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}$ .

The display will show the equation,

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

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)

The function $x\sin x$ tends to $0$ as $x$ tends to $0$ from both the sides. So, the value of $\underset{x\to 0}{\lim }\left(x\sin x\right)$ graphically is $0$ .

Check the limit by sandwich theorem.

As the value of function $\sin x$ lies between $-1$ and $1$ . So, the value of function $x\sin x$ lies between $-x$ and $x$ .

  $-x\le x\sin x\le x$

Calculate the limit of $-x$ and $x$ .

  $\underset{x\to 0}{\lim }\left(-x\right)=0$

And,

  $\underset{x\to 0}{\lim }\left(x\right)=0$

So, the limit $\underset{x\to 0}{\lim }\left(x\sin x\right)$ is $0$ .

Therefore, the value of $\underset{x\to 0}{\lim }\left(x\sin x\right)$ is $0$ .