Answer:
Option D
Explanation:
Plan $\lim_{x \rightarrow 0}\frac{\sin x}{x}=1$
Given, $\lim_{x \rightarrow 1}\left\{\frac{\sin (x-1)+a(1-x)}{(x-1)+\sin (x-1)}\right\}^{\frac{(1+\sqrt{x})(1-\sqrt{x})}{(1-\sqrt{x})}}=\frac{1}{4}$
$\lim_{x \rightarrow 1}\left\{\frac{\frac{\sin(x-1)}{(x-1)}-a}{1+\frac{\sin(x-1)}{(x-1)}}\right\}^{1+\sqrt{x}}=\frac{1}{4}$
$\Rightarrow$ $\left(\frac{1-a}{2}\right)^{2}=\frac{1}{4}$
$\Rightarrow$ (a-1)2=1
$\Rightarrow$ a=2 or 0
But for a=2 base of above limit approaches -1/2 and exponent approaches to 2 and since base cannot be negative. Hence limit does not exist
