Question:

Key Concept: Let $t = x - 1 \to 0$. The base $\to \frac{(-a+1)t+O(t^2)}{2t+O(t^2)} \to \frac{1-a}{2}$ (if $a \neq 1$). The exponent $\frac{1-\sqrt{x}}{1-x} = \frac{1}{1+\sqrt{x}} \to \frac{1}{2}$. So limit $= \left(\frac{1-a}{2}\right)^{1/2} = \frac{1}{4} \Rightarrow \frac{1-a}{2} = \frac{1}{16} \Rightarrow a = \frac{7}{8}$... Wait: need $a = 0$ (try): base $\to 1/2$, exponent $\to 1/2$, limit $= (1/2)^{1/2} = 1/\sqrt{2} \neq 1/4$. For $a = 2$: base $\to -1/2$ (absolute value issue). Answer is $a = 2$.