The largest value of the non-negative integer a for which `lim_(xrarr1){(-ax+sin(x-1)+a)/(x+sin(x-1)-1)}^((1-x)/(1-sqrtx)}= 1/4

Answer

`(2)`

Explanation

`underset(xto1)lim{(-ax+sin(x-1)+a)/(x+sin(x-1)-1)}^((1-x)/(1-sqrt(x)))=(1)/(4)`
`implies" "underset(xto1)lim{((sin(x-1))/((x-1))-a)/((sin(x-1))/((x-1))+1)}^(1+sqrt(x))=(1)/(4)`
`implies" "((1-a)/(2))^(1)=(1)/(4)`
`implies" "a=0,a=2`