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Q. $\displaystyle \lim_{x \to 0}$ $\frac{\left|sin\,x\right|}{x}$ is

Limits and Derivatives

Solution:

We have, $\displaystyle \lim_{x \to 0}$ $\frac{\left|sin\,x\right|}{x}$
$L.H.L$ $=\displaystyle \lim_{x \to 0^-}$ $\frac{\left|sin\,x\right|}{x}$
$=\displaystyle \lim_{h \to 0}$ $-\left(\frac{sin\,h}{h}\right)$
$=\left(-1\right)\times1=-1$
$R.H.L.$ $=\displaystyle \lim_{x \to 0^+}$ $\frac{\left|sin\,x\right|}{x}$
$=\displaystyle \lim_{h \to 0}$ $\frac{sin\,h}{h}=1$
Since $L.H.L. \ne R.H.L.$
$\therefore $ Limit does not exist