limx→0tan−1x−sin−1xx3is equal to

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Question

{lim}_{x \to 0} \frac{t a n^{- 1} x - s i n^{- 1} x}{x^{3}} is equal to

Options:

A . 1/2

B . -1/2

C . 1

D . -1

Answer: Option B
:
B

l i m x \to 0 \frac{t a n^{- 1} x - s i n^{- 1} x}{x^{3}} (\frac{0}{0} f o r m) l i m x \to 0 \frac{\frac{1}{1 + x^{2}} - \frac{1}{\sqrt{1 - x^{2}}}}{3 x^{2}} l i m x \to 0 \frac{\sqrt{1 - x^{2}} - (1 + x^{2})}{3 x^{2} (1 + x^{2}) \sqrt{1 - x^{2}}} l i m x \to 0 \frac{(1 - x^{2}) - (1 + x^{2})^{2}}{3 x^{2} (1 + x^{2}) \sqrt{1 - x^{2}} [\sqrt{1 - x^{2}} + (1 + x^{2})]} = - \frac{3}{6} = - \frac{1}{2}

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