Lim x- 0 sin -1 x- tan -1 x x 3 -

The correct option is D 16 lim _ x→ 0 sin ^ 1 x tan ^ 1 x #160; x ^ 3 #160; ⇒ 1 (1 x ^ 2 ) ^ 1 / 2 #160; 1 (1+ x ^ 2 ) ...

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Byju's Answer

Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

lim x→ 0 sin ...

Question

$lim x \to 0 \frac{{sin}^{- 1} x - {tan}^{- 1} x}{x^{3}} =$

2

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\frac{1}{6}

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Solution

The correct option is D $\frac{1}{6}$
${lim}_{x \to 0} \frac{{sin}^{- 1} x - {tan}^{- 1} x}{x^{3}} \Rightarrow \frac{\frac{1}{{(1 - x^{2})}^{\frac{1}{2}}} - \frac{1}{(1 + x^{2})}}{3 x^{2}} \Rightarrow \frac{{(1 - x^{2})}^{\frac{- 1}{2}} - {(1 + x^{2})}^{- 1}}{3 x^{2}}$
Using LHR.
$\Rightarrow \frac{{\frac{1}{2} (1 - x^{2})}^{\frac{- 3}{2}} \times 2 x + {(1 + x^{2})}^{- 2} \times 2 x}{6 x} \Rightarrow \frac{{x (1 - x^{2})}^{\frac{- 3}{2}} + {(1 + x^{2})}^{- 2} \times 2 x}{6 x}$

$= \frac{1}{6}$

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