The value of `f(0),` so that the function `f(x)=(sqrt(a^2-a x+x^2)-sqrt(a^2+a x+x^2))/(sqrt(a+x)-sqrt(a-x))` becomes continuous for all `x ,` given by `a^(3/2)` (b) `a^(1/2)` (c) `-a^(1/2)` (d) `-a^(3/2)`

Options

A. `a^(3//2)`
B. `a^(1//2)`
C. `-a^(1//2)`
D. `-a^(3//2)`

Answer

C

Explanation

`f(x)=(sqrt(a^(2)-ax+x^(2))-sqrt(a^(2)+ax+x^(2)))/(sqrt(a+x)-sqrt(a-x)) xx (sqrt(a^(2)-ax+x^(2))+sqrt(a^(2)+ax+x^(2)))/(sqrt(a^(2)-ax+x^(2))+sqrt(a^(2)+ax+x^(2)))xx(sqrt(a+x)+sqrt(a-x))/(sqrt(a+x)+sqrt(a-x))`
`rArr lim_(x to 0)f(x)=lim_(x to 0)(-2ax(sqrt(a+x)+sqrt(a-x)))/(2x(sqrt(a^(2)-ax+x^(2))+sqrt(a^(2)+ax+x^(2))))`
`=(-a (2sqrt(a)))/(a+a)=-sqrt(a)`