lim _{x rightarrow 1} frac{sqrt{x}-1}{(cos ^{ | vedclass

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Question

(A) Let $x = \cos 	heta$. As $x ightarrow 1$,$	heta ightarrow 0$. Substituting this into the limit: $\lim _{	heta ightarrow 0} \frac{\sqrt{\c

Vedclass · Accepted Answer

$-\frac{1}{4}$

Vedclass · Answer

(A) Let $x = \cos 	heta$. As $x ightarrow 1$,$	heta ightarrow 0$. Substituting this into the limit: $\lim _{	heta ightarrow 0} \frac{\sqrt{\cos 	heta}-1}{	heta^2}$ Multiply the numerator and denominator by $(\sqrt{\cos 	heta}+1)$: $= \lim _{	heta ightarrow 0} \frac{\cos 	heta - 1}{	heta^2(\sqrt{\cos 	heta}+1)}$ Using the identity $\cos 	heta - 1 = -2 \sin^2(\frac{	heta}{2})$: $= \lim _{	heta ightarrow 0} \frac{-2 \sin^2(\frac{	heta}{2})}{	heta^2(\sqrt{\cos 	heta}+1)}$ $= \lim _{	heta}$ ${ightarrow 0} -2 \cdot \left(\frac{\sin(\frac{	heta}{2})}{\frac{	heta}{2} \cdot 2}ight)^2 \cdot \frac{1}{\sqrt{\cos 	heta}+1}$ $= -2 \cdot \frac{1}{4} \cdot \frac{1}{1+1} = -2 \cdot \frac{1}{4} \cdot \frac{1}{2} = -\frac{1}{4}$.

$\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{(\cos ^{-1} x)^2} = $

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