The largest value of the non-negative integer $a$ for which   $\displaystyle \lim_{x \rightarrow 1} \displaystyle \left \{ \dfrac{-ax + \sin (x-1)+ a}{x+\sin (x-1)-1} \right \}^{\dfrac{1-x}{1-\sqrt{x}}} = \dfrac{1}{4}$ is ................

$\displaystyle \lim_{x \rightarrow 1} \left ( \displaystyle \frac{-ax + sin (x-1)+a}{x+sin (x-1)-1} \right )^{\displaystyle \frac{1-x}{1-\sqrt{x}}} = \displaystyle \frac{1}{4}$

$\displaystyle \lim_{x \rightarrow 1} \displaystyle \left ( \displaystyle \frac{\displaystyle \frac{sin (x-1)}{(x-1)}-a}{\displaystyle \frac{sin(x-1)}{(x-1)}+1}\right )^{1+\sqrt{x}} = \displaystyle \frac{1}{4} \Rightarrow \left (\displaystyle \frac{1-a}{2} \right)^2 =\displaystyle \frac{1}{4}$

$\Rightarrow a = 0, a= 2$

they asked for non negative intger so a =2