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  4. The value of displaystyle lim x arrow 1- (1-√x/( cos -1 x)2) is

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Q. The value of $\displaystyle\lim _{x \rightarrow 1^{-}} \frac{1-\sqrt{x}}{\left(\cos ^{-1} x\right)^{2}}$ is

Limits and Derivatives

Solution:

We have
$\displaystyle\lim _{x \rightarrow 1} \frac{1-\sqrt{x}}{\left(\cos ^{-1} x\right)^{2}}$
$=\displaystyle\lim _{x \rightarrow 1} \frac{(1-\sqrt{x})(1+\sqrt{x})}{\left(\cos ^{-1} x\right)^{2}(1+\sqrt{x})}$
$=\displaystyle\lim _{x \rightarrow 1} \frac{1-x}{\left(\cos ^{-1} x\right)^{2}(1+\sqrt{x})}$
$=\displaystyle\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta^{2}(1+\sqrt{\cos \theta})} 1$,
where $x=\cos \theta$
$[\because x \rightarrow 1$ or $\cos \theta \rightarrow 1$ or $\theta \rightarrow 0]$
$=\displaystyle\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta^{2}} \frac{1}{(1+\sqrt{\cos \theta})}$
$=\displaystyle\lim _{\theta \rightarrow 0} \frac{2 \sin ^{2} \frac{\theta}{2}}{\theta^{2}}\left(\frac{1}{1+\sqrt{\cos \theta}}\right)$
$=\frac{1}{2} \displaystyle\lim _{\theta \rightarrow 0}\left(\frac{\sin \frac{\theta}{2}}{\frac{\theta}{2}}\right)^{2} \frac{1}{(1+\sqrt{\cos \theta})}$
$=\frac{1}{2}(1)^{2} \frac{1}{(1+1)}=\frac{1}{4}$