What is \(\displaystyle\lim_{x\to 0}\dfrac{\sin^{2}x}{x|x|}\) equal to?
Limit does not exist
For \(x\to 0^{+}\), \(|x|=x\), so the expression equals \(\dfrac{\sin^{2}x}{x^{2}}\to 1\). For \(x\to 0^{-}\), \(|x|=-x\), so the expression equals \(-\dfrac{\sin^{2}x}{x^{2}}\to -1\). Since the left-hand and right-hand limits differ, the limit does not exist.
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