Question

Evaluate:
$\underset{x\to \frac{1}{\sqrt{2}}}{lim}$ $\frac{x-\cos \left({\sin }^{-1}x\right)}{1-\tan \left({\sin }^{-1}x\right)}$

Answer 1

$\underset{x\to \frac{1}{\sqrt{2}}}{lim}\frac{x-\cos \left({\text{sin}}^{-1}x\right)}{1-\tan \left({\text{sin}}^{-1}x\right)}=\underset{x\to \frac{1}{\sqrt{2}}}{lim}\frac{1+\sin \left({\text{sin}}^{-1}x\right)\frac{1}{\sqrt{1-x^2}}}{-{\text{sec}}^2\left({\text{sin}}^{-1}x\right)\frac{1}{\sqrt{1-x^2}}}=\frac{1+1}{-2\left(\sqrt{2}\right)}=-\frac{1}{\sqrt{2}}$