The largets value of non negative integer for which `lim_(x->1){(-a x+sin(x-1)+a)/(x+sin(x-1)-1)}^((1-x)/(1-sqrt(x)))=1/4`

Options

A. 2
B. 3
C. 4
D. 5

Answer

A

Explanation

We have ,
`lim_(alphato1){(-ax+sin(x-1)+a)/(x+sin(x-1)-1)}^((1-x)/(1-sqrt(x)))=(1)/(4)`
`lim_(alphato1){((sin(x-1))/(x-1)-a)/(1+(sin(x-1))/(x-1))}^(1+sqrt(x))=(1)/(4)`
`rArr ((1-a)/(2))^2=(1)/(4)`
`rArr a^2=2a=0rArra=0,2`.