\(\lim\limits_{x\to 1}\frac{sin^{-1}(1-√x)}{1-x}\)
= \(\lim\limits_{ x \to 1}\cfrac{sin^{-1}(\frac{(1-\sqrt x)(1+\sqrt x)}{1+\sqrt x})}{1-x}\)
= \(\lim\limits_{ x \to 1}\cfrac{sin^{-1}(\frac{(1- x)}{1+\sqrt x})}{1-x}\)
= \(\lim\limits_{ x \to 1}\cfrac{(\frac{1- x}{1+\sqrt x})+\frac1{2\times3}(\frac{1-x}{1+√ x})^3 + \frac{1.3}{2\times4\times5}(\frac{1-x}{1+\sqrt x})^5+....}{1-x}\)
(∵ sin-1x = x + 1/2 x3/3 + 1.3/2.4 x5/5 + ....)
= \(\lim\limits_{ x \to 1}(\frac1{1+\sqrt x}+\frac1{6}\frac{(1-x)^2}{(1+\sqrt x)^3}+\frac3{40}\frac{(1-x^4)}{(1+\sqrt x)^5+...})\)
= \(\frac1{1+\sqrt1}+0\)
= 1/2