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

Given expression,

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

$\because {lim}_x\to a\frac{\sin 0}{0}=\mathrm{1....}\left(i\right)$

A

From (i) we can write above expression as-

$\underset{x\to 1}{lim}{\left(\frac{\frac{sin(x-1)}{(x-1)}-a}{\frac{sin(x-1)}{(x-1)}+1}\right)}^{1+\sqrt{x}}=\frac{1}{4}\Rightarrow {\left(\frac{1-a}{2}\right)}^2=\frac{1}{4}$

$\frac{1-a}{2}=\pm \frac{1}{2}$

$\Rightarrow a=0,a=2$
Hence, non negative integer is $a=2$