Question

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

• A. $0$
• B. $1$
• C. $-1$
• D. Does not exist

Answer 1

The correct option is D Does not exist

We know, $|x|=\{\begin{array}{c}-x,x<0\\ x,x>0\end{array}$
Thus,
L.H.L.$=\underset{x\to 0^{-}}{lim}\frac{\sin x}{|x|}=\underset{x\to 0^{-}}{lim}\frac{\sin x}{-x}=-\underset{x\to 0^{-}}{lim}\frac{\sin x}{x}=-1$

And,

R.H.L. $=\underset{x\to 0^{+}}{lim}\frac{\sin x}{|x|}=\underset{x\to 0^{+}}{lim}\frac{\sin x}{x}=\underset{x\to 0^{+}}{lim}\frac{\sin x}{x}=1$

Clearly L.H.L.$\ne$ R.H.L. $\Rightarrow$ limit does not exist.