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What is the value of $\lim _{x \rightarrow 0} \frac{\sin |x|}{x}$ ?
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Limit does not exist
$\lim _{\mathrm{x} \rightarrow 0} \frac{\sin |\mathrm{x}|}{\mathrm{x}}, \mathrm{LHL}$ is limit when $\mathrm{x} < 0$
$\mathrm{LHL}=\lim _{\mathrm{x} \rightarrow 0} \frac{\sin (-\mathrm{x})}{\mathrm{x}}=-\lim _{\mathrm{x} \rightarrow 0} \frac{\sin \mathrm{x}}{\mathrm{x}}=-1$
RHL is limit when $\mathrm{x}>0$
$\mathrm{RHL}=\lim _{\mathrm{x} \rightarrow 0} \frac{\sin (\mathrm{x})}{\mathrm{x}}=1$
So, LHL $\neq$ RHL Hence, Limit does not exist.
$\mathrm{LHL}=\lim _{\mathrm{x} \rightarrow 0} \frac{\sin (-\mathrm{x})}{\mathrm{x}}=-\lim _{\mathrm{x} \rightarrow 0} \frac{\sin \mathrm{x}}{\mathrm{x}}=-1$
RHL is limit when $\mathrm{x}>0$
$\mathrm{RHL}=\lim _{\mathrm{x} \rightarrow 0} \frac{\sin (\mathrm{x})}{\mathrm{x}}=1$
So, LHL $\neq$ RHL Hence, Limit does not exist.
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