Answer
$\displaystyle\lim_{x\rightarrow 0^+} \dfrac{\sin x}{|x|}=-1$
$\displaystyle\lim_{x\rightarrow 0^-} \dfrac{\sin x}{|x|}=1$
The limit in 0 does not exist
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{x\rightarrow \pm0} \dfrac{\sin x}{|x|}$
Compute $\dfrac{\sin x}{|x|}$ for values of $x$ approaching 0 from the left and from the right side:
$\dfrac{\sin (-0.1)}{|-0.1|}\approx -0.99833417$
$\dfrac{\sin (-0.01)}{|-0.01|}\approx -0.99998333$
$\dfrac{\sin (-0.001)}{|-0.001|}\approx -0.99999983$
$\dfrac{\sin (0.1)}{|0.1|}\approx 0.99833417$
$\dfrac{\sin (0.01)}{|0.01|}\approx 0.99998333$
$\dfrac{\sin (0.001)}{|0.001|}\approx 0.99999983$
Therefore we got:
$\displaystyle\lim_{x\rightarrow 0^+} \dfrac{\sin x}{|x|}=-1$
$\displaystyle\lim_{x\rightarrow 0^-} \dfrac{\sin x}{|x|}=1$
As the left hand limit and the right hand limit are not equal, the limit of the function in 0 does not exist.
