The largest value of the non-negative integer a for which $\underset{\alpha \to 1}{lim} {\left\{\frac{-ax+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4},$ is 

Solution:

We have.

        $\underset{\alpha \to 1}{lim} {\left\{\frac{-ax+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$

$\Rightarrow \underset{\alpha \to 1}{lim} {\left\{\frac{\frac{\sin (x-1)}{x-1}-a}{1+\frac{\sin (x-1)}{x-1}}\right\}}^{1+\sqrt{x}}=\frac{1}{4}$

$\begin{array}{l}\Rightarrow {\left(\frac{1-a}{2}\right)}^2=\frac{1}{4}\\ \Rightarrow a^2-2a=0\Rightarrow a=0,2\end{array}$