\(\lim\limits_{x\to 1} \left\{ \frac{-ax + \sin(x-1) + a}{x + \sin(x-1)-1}\right\}^{\frac{1-x}{1-\sqrt x}}\)
\(=\left\{\lim\limits_{x\to 1} \frac{-ax + \sin(x-1) + a}{x + \sin(x-1)-1}\right\}^{\lim\limits_{x\to 1}\frac{1-x}{1-\sqrt x}}\) \(\left(\frac 00-cases\right)\)
\(=\left\{\lim\limits_{x\to 1} \frac{-a + \cos(x-1)}{1 + \cos(x-1)}\right\}^{\lim\limits_{x\to 1}\frac{-1}{\frac{-1}{2\sqrt x}}}\) (By applying D.L.H. Rule)
\(= \left\{\frac{-a+1}2\right\}^2\)
\(= \frac{(1-a)^2}4\)
\(= \frac{(a-1)^2}4\)
