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Question: \(\lim _ { x \rightarrow - 1 }\) \(\frac{\sqrt{\pi} - \sqrt{(\cos^{- 1}x)}}{\sqrt{(x + 1)}}\) is giv...

lim⁡x→−1\lim _ { x \rightarrow - 1 } π−(cos⁡−1x)(x+1)\frac{\sqrt{\pi} - \sqrt{(\cos^{- 1}x)}}{\sqrt{(x + 1)}} is given by-

A

1/π\sqrt{\pi}

B

1/(2π)1/\sqrt{(2\pi)}

C

1

D

0

Answer

1/(2π)1/\sqrt{(2\pi)}

Explanation

Solution

Let cos–1 x = y

Q x ® –1, y ® cos–1 (–1) Ž y ® p

lim⁡y→π\lim_{y \rightarrow \pi}(π+yπ+y)\left( \frac{\sqrt{\pi} + \sqrt{y}}{\sqrt{\pi} + \sqrt{y}} \right)

lim⁡y→π\lim _ { y \rightarrow \pi } π−y2.cos⁡y2(π+y)\frac{\pi - y}{\sqrt{2}.\cos\frac{y}{2}\left( \sqrt{\pi} + \sqrt{y} \right)} put y = p + h

lim⁡h→π\lim_{h \rightarrow \pi} −h2.(−sin⁡h2)(h2)[π+π+h].h2\frac{- h}{\sqrt{2}.\frac{\left( - \sin\frac{h}{2} \right)}{\left( \frac{h}{2} \right)}\left\lbrack \sqrt{\pi} + \sqrt{\pi + h} \right\rbrack.\frac{h}{2}}= 12π\frac{1}{\sqrt{2\pi}}