`lim_(xtoa){[((a^(1//2)+x^(1//2)),(a^(1//4)-x^(1//4)))^(-1)-(2(ax)^(1//4))/(x^(3//4)-a^(1//4)x^(1//2)+a^(1//2)x^(1//4)-a^(3//4))]^(-1)-sqrt(2)^(log_(4)a)}^(8)` is

Options

A. a
B. `a^(3//4)`
C. `a^2`
D. `sqrt(a)`

Answer

C

Explanation

We have,
`((sqrt(a)+sqrt(a))/(a^(1//4-x^(1//4))))^-1-(2(ax)^(1//4))/(x^(3//4)-a^(1//4)x^(1//2)+a^(1//2)-a^(3//4))`
`=(a^(1//4)-x^(1//4))/(sqrt(x)+sqrt(a))-(2(ax)^(1//4))/((x^(1//4)-a^(1//4))(x^(1//2)+a^(1//2)))`
`=(-(x^(1//4)-a^(1//4))^2-2(ax)^(1//4))/((x^(1//4)-a^(1//4))(x^(1//2)+a^(1//2)))=-(1)/(x^(1//4)-a^(1//4))`
and, `(sqrt(2))^(log4a)=2^(1//2log4a)=2^(1//4log2a)=2^(log2)a^(1//4)=a^(1//4)`
` therefore` Required limit `=lim_(xtoa){-(x^(1//4)-a^(//4))-a^(1//4)}^8`
` = {-a^(1//4)}^8=a^2`.