lim x → 0 t a n − 1 x − s i n − 1 x x 3 is equal to Options: A . &nbsp 1/2 B . &nbsp -1/2 C . &nbsp 1 D . &nbsp -1

Answer: Option B : B l i m x → 0 t a n − 1 x − s i n − 1 x x 3 ( 0 0 f o r m ) l i m x → 0 1 1 + x 2 − 1 √ 1 − x 2 3 x 2 l i m x → 0 √ 1 − x 2 − ( 1 + x 2 ) 3 x 2 ( 1 + x 2 ) √ 1 − x 2 l i m x → 0 ( 1 − x 2 ) − ( 1 + x 2 ) 2 3 x 2 ( 1 + x 2 ) √ 1 − x 2 [ √ 1 − x 2 + ( 1 + x 2 ) ] = − 3 6 = − 1 2

limx→0tan−1x−sin−1xx3is equal to

Discussion Forum : Limits Continuity And Differentiability

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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Discussion Forum : Limits Continuity And Differentiability

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