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  4. displaystyle lim x arrow (1/√2) ( sin ( cos -1 x)-x/1- tan ( cos -1 x)) is equal to :

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

JEE MainJEE Main 2022Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\cos ^{-1} x\right)-x}{1-\tan \left(\cos ^{-1} x\right)}$ $\displaystyle\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\sin ^{-1} \sqrt{1-x^{2}}\right)-x}{1-\tan \left(\tan ^{-1}\left(\frac{\sqrt{1-x^{2}}}{x}\right)\right)}$
$\displaystyle\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{\sqrt{1-x^{2}}-x}{1-\left(\frac{\sqrt{1-x^{2}}}{x}\right)}$ $\displaystyle\lim _{x \rightarrow \frac{1}{\sqrt{2}}}(-x)=-\frac{1}{\sqrt{2}}$