Question

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

Answer 1

${lim}_{x\to 1}{\left\{\frac{-ax+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}$
${lim}_{x\to 1}{\left\{\frac{a(1-x)+\sin (x-1)}{(x-1)+\sin (x-1)}\right\}}^{1+\sqrt{x}}$
${lim}_{x\to 1}{\left\{\frac{-a+\frac{\sin (x-1)}{x-1}}{1+\frac{\sin (x-1)}{x-1}}\right\}}^{(1+\sqrt{x})}$
${\left\{\frac{-a+1}{2}\right\}}^2=\frac{1}{4}$
$\frac{-a+1}{2}=\pm \frac{1}{2}$
$–a+1=\pm 1\Rightarrow a=0,a=2$
But at $a=2$, base becomes negative
$\therefore a=0$