Question

Evaluate: $\underset{x\to 0}{\lim }\frac{\sin \left|x\right|}{x}$

• A. $1$
• B. $0$
• C. positive infinity
• D. does not exist

Answer 1

The correct option is D

does not exist

Explanation for the correct option:

Step-1 Finding the left hand limit:

Consider the given Equation as,

$I=\underset{x\to 0}{\lim }\frac{\sin \left|x\right|}{x}$

Find the the LHS, when $x\to 0-h\mathbf{}\left[\text{Where,}h>0\right]$

$$\begin{gathered} I=\underset{h\to 0}{\lim }\frac{\sin \left|0-h\right|}{0-h} \\ I=\underset{h\to 0}{\lim }\frac{\sin \left|-h\right|}{-h} \\ I=\underset{h\to 0}{\lim }\frac{\sin h}{-h} \\ I=-1 \end{gathered}$$

Step-2 Finding the right hand limit:

Find the the RHS, when $x\to 0+h\left[\text{Where,}h>0\right]$

$$\begin{gathered} I=\underset{h\to 0}{\lim }\frac{\sin \left|0+h\right|}{0+h} \\ I=\underset{h\to 0}{\lim }\frac{\sin \left|h\right|}{h} \\ I=\underset{h\to 0}{\lim }\frac{\sin h}{h} \\ I=1 \end{gathered}$$

Since, $LHL\ne RHL$

Hence, The limit does not exist.

Therefore, the correct answer is option D.