A siny limit

$\begin{aligned} \frac 14 & = \lim_{x \to 1} \left(\frac {-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right)^{\frac {1-x}{1-\sqrt x}} \\ & = \lim_{x \to 1} \left(\frac {-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right)^{\frac {(1-\sqrt x)(1+\sqrt x)}{1-\sqrt x}} \\ & = \lim_{x \to 1} \left(\frac {-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right)^{1+\sqrt x} \\ & = \left(\lim_{x \to 1} \frac {-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right)^2 \\ \implies \pm \frac 12 & = \lim_{x \to 1} \frac {-ax+\sin(x-1)+a}{x+\sin(x-1)-1} & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \lim_{x \to 1} \frac {-a+\cos (x-1)}{1+\cos (x-1)} & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x \\ & = \frac {-a+1}2 \end{aligned}$

$\implies \begin{cases} -a+1 = 1 & \implies a = 0 & \color{#D61F06} \text{rejected} \\ -a+1 = -1 & \implies a = \boxed{2} & \color{#3D99F6} \text{accepted} \end{cases}$