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

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

Limits and Derivatives

Solution:

Let $\sin ^{-1} x=\theta$.
Then $x=\sin \theta$.
Now, $x \rightarrow \frac{1}{\sqrt{2}} $
$\Rightarrow \sin \theta \rightarrow \frac{1}{\sqrt{2}}$
$ \Rightarrow \theta \rightarrow \frac{\pi}{4}$
$\therefore \displaystyle\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{x-\cos \left(\sin ^{-1} x\right)}{1-\tan \left(\sin ^{-1} x\right)}$
$=\displaystyle\lim _{\theta \rightarrow \frac{\pi}{4}} \frac{\sin \theta-\cos \theta}{1-\tan \theta}$
$=\displaystyle\lim _{\theta \rightarrow \frac{\pi}{4}} \frac{(\sin \theta-\cos \theta)}{(\cos \theta-\sin \theta)} \cos \theta$
$=\displaystyle\lim _{\theta \rightarrow \frac{\pi}{4}}-\cos \theta=-\frac{1}{\sqrt{2}}$