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Q. $\displaystyle \lim _{x \rightarrow 0} \sqrt{\frac{x-\sin x}{x+\sin ^{2} x}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle \lim _{x \rightarrow 0} \sqrt{\frac{x-\sin x}{x+\sin ^{2} x}}$
$=\displaystyle \lim _{x \rightarrow 0} \sqrt{\frac{1-\frac{\sin x}{x}}{1+\frac{\sin ^{2} x}{x}}}$
$=\displaystyle \lim _{x \rightarrow 0} \sqrt{\frac{1-\frac{\sin x}{x}}{1+\left(\frac{\sin x}{x}\right) \sin x}}$
$=\sqrt{\frac{1-1}{1+1 \times 0}}=0$