Chapter 1.1, Problem 65E

To determine

To find: The limit graphically and use the Squeeze Theorem to confirm the answer.

Answer to Problem 65E

The required answer is 0.

Explanation of Solution

Given information: The expression is $\underset{x\to 0}{\lim }[x\sin x]$

Graph:

The function is graphed, and it is discovered that the specified limit is 0.

  

A real $x$ , be aware of that,

  $-1\le \sin x\le 1$

Multiply the inequality's components by x.

  $-x\le x\sin x\le x$

Here assume $x>0$ , The inequalities are reserved if $x<0$ and the expression is unaffected.

Notice that,

  $\begin{array}{c}\underset{x\to 0}{\lim }(-x)=0\\ \underset{x\to 0}{\lim }x=0\end{array}$

Along with,

• $-x\le x\sin x$;
• $\sin x\le x$; and
• $\begin{array}{c}\underset{x\to 0}{\lim }(-x)=\underset{x\to 0}{\lim }x\\ =0\end{array}$

Next, to use the sandwich theory,

  $\underset{x\to 0}{\lim }[x\sin x]=0$

Therefore, the required value of $\underset{x\to 0}{\lim }[x\sin x]=0$