Answer
$\displaystyle\lim_{x\rightarrow 0^-} \dfrac{x-\sin |x|}{x^3}=\infty$
$\displaystyle\lim_{x\rightarrow 0^+} \dfrac{x-\sin |x|}{x^3}\approx 0.167$
The limit does not exist
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{x\rightarrow \pm0} \dfrac{x-\sin |x|}{x^3}$
Graph the function:
Therefore we get:
$\displaystyle\lim_{x\rightarrow 0^-} \dfrac{x-\sin |x|}{x^3}=\infty$
$\displaystyle\lim_{x\rightarrow 0^+} \dfrac{x-\sin |x|}{x^3}\approx 0.167$
As the left hand limit and the right hand limit are not equal, the limit of the function in 0 does not exist. The function tends to $\infty$ when $x\rightarrow 0+$.
