Discussion Forum

The largest value of the non-negative integer a for which $\lim_{x \rightarrow 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:

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