Question

$\displaystyle\lim_{x\to0} \sqrt{\frac{x-\sin x}{x+\sin^{2}x}} $ is equal to

• A. 1
• B. 0
• C. $\infty$
• D. None of these

Answer 1

Answer: (B)

$\displaystyle\lim_{x\to0} \sqrt{\frac{x-\sin x}{x+\sin^{2}x}} = \displaystyle\lim_{x\to0} \sqrt{\frac{1- \frac{\sin x}{x}}{1+ \frac{\sin^{2}x}{x}}} $
$=\displaystyle\lim_{x\to0} \sqrt{\frac{1- \frac{\sin x}{x}}{1+ \left(\frac{\sin x}{x}\right)\sin x}} = \sqrt{\frac{1-1}{1+1\times0}} = 0 $