Question

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Answer 1

Answer:

(4) 1/4 is correct.

Step-by-step explanation:

$Put \cos ^{-1} x = y \: and \: x \rightarrow 1^{-} \Rightarrow y \rightarrow 0\\</p><p>\operatorname{Lim}_{ x \rightarrow 1} \frac{1-\sqrt{ x }}{\left(\cos ^{-1} x \right)^{2}}=\operatorname{Lim}_{ y \rightarrow 0} \frac{1-\sqrt{\cos y }}{ y ^{2}}\\</p><p> Now rationalizing numerator \\</p><p>\operatorname{Lim}_{y \rightarrow 0} \frac{1-\cos y}{y^{2}(1+\sqrt{\cos y})}\\</p><p>=\lim _{y \rightarrow 0} \frac{1-\cos y}{y^{2}} \operatorname{Lim}_{y \rightarrow 0} \frac{1}{(1+\sqrt{\cos y})}\\</p><p>=\frac{1}{2} \operatorname{Lim}_{y \rightarrow 0} \frac{2 \sin ^{2} \frac{y}{2}}{\left(\frac{y}{2}\right)^{2}} \times \frac{1}{4}=\frac{1}{4}</p><p>$

Answered by Gauthmath

Answer 2

Option (4.) is correct

Solution is given in the attachment