\(\lim\limits_{x\to 1}\frac{sin^{-1}(1-√x)}{1-x}\)

= \(\lim\limits_{ x \to 1}\cfrac{sin^{-1}(\frac{(1-\sqrt x)(1+\sqrt x)}{1+\sqrt x})}{1-x}\)

= \(\lim\limits_{ x \to 1}\cfrac{sin^{-1}(\frac{(1- x)}{1+\sqrt x})}{1-x}\)

= \(\lim\limits_{ x \to 1}\cfrac{(\frac{1- x}{1+\sqrt x})+\frac1{2\times3}(\frac{1-x}{1+√ x})^3 + \frac{1.3}{2\times4\times5}(\frac{1-x}{1+\sqrt x})^5+....}{1-x}\)

**(∵ sin**^{-1}x = x + 1/2 x^{3}/3 + 1.3/2.4 x^{5}/5 + ....)

= \(\lim\limits_{ x \to 1}(\frac1{1+\sqrt x}+\frac1{6}\frac{(1-x)^2}{(1+\sqrt x)^3}+\frac3{40}\frac{(1-x^4)}{(1+\sqrt x)^5+...})\)

= \(\frac1{1+\sqrt1}+0\)

= 1/2