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) `a^(3/2)` (b) `a^(1/2)` (c) `-a^(1/2)` (d) `-a^(3/2)`

Explanation

We have given

`f(x)=(sqrt(a^2-a x+x^2)-sqrt(a^2+a x+x^2))/(sqrt(a+x)-sqrt(a-x))`

f(x) is continuous for all x so it is continuous at x=0

`f(0)=lim_(x->0)f(x)`

`=lim_(x->0) (sqrt(a^2-a x+x^2)-sqrt(a^2+a x+x^2))/(sqrt(a+x)-sqrt(a-x))`

`=lim_(x->0)((sqrt(a^2-a x+x^2)-sqrt(a^2+a x+x^2))/(sqrt(a+x)-sqrt(a-x)))\times(sqrt(a+x)+sqrt(a-x))/(sqrt(a+x)+sqrt(a-x))\times(sqrt(a^2-a x+x^2)+sqrt(a^2+a x+x^2))/(sqrt(a^2-a x+x^2)+sqrt(a^2+a x+x^2))`

`=lim_(x->0)(-2ax)/(2a)lim_(x->0)((sqrt(a+x)+sqrt(a−x))/(sqrt(a^2−ax+x^2)
+ sqrt(a^2+ax+x^2)))`

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