## Calculus (3rd Edition)

$\displaystyle\lim_{x\rightarrow 0^+} \dfrac{\sin x}{|x|}=-1$ $\displaystyle\lim_{x\rightarrow 0^-} \dfrac{\sin x}{|x|}=1$ The limit in 0 does not exist
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.