Chapter 2.1, Problem 72E

To determine

To check: Whether the statement for the limit $\underset{x\to 0}{\lim }\left(\frac{x+\sin x}{x}\right)=2$ is true or not.

Answer to Problem 72E

The given statement is true.

Explanation of Solution

Given information: The given statement is “ $\underset{x\to 0}{\lim }\left(\frac{x+\sin x}{x}\right)=2$ ”.

Simplify the limit.

  $\begin{array}{c}\underset{x\to 0}{\lim }\left(\frac{x+\sin x}{x}\right)=\underset{x\to 0}{\lim }\left(1+\frac{\sin x}{x}\right)\\ =\underset{x\to 0}{\lim }1+\underset{x\to 0}{\lim }\left(\frac{\sin x}{x}\right)\\ =1+1\\ =2\end{array}$

Therefore, the given statement is true.