The largest value of the non-negative integer a for which limα→1 −ax+sin⁡(x−1)+ax+sin⁡(x−1)−11−x1−x=14, is

We have.

$lim_{α \to 1} {\{\frac{- a x + \sin (x - 1) + a}{x + \sin (x - 1) - 1}\}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4}$

$\Rightarrow lim_{α \to 1} {\{\frac{\frac{\sin (x - 1)}{x - 1} - a}{1 + \frac{\sin (x - 1)}{x - 1}}\}}^{1 + \sqrt{x}} = \frac{1}{4}$

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

The largest value of the non-negative integer a for which $lim_{α \to 1} {\{\frac{- a x + \sin (x - 1) + a}{x + \sin (x - 1) - 1}\}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4},$ is