Question

Lim x → ∞{(x+a)(x+b)}^1/2 -x) equals?

Answer 1

`(a+b)/2`

Explanation:

`sqrt((x+a)(x+b))-x = ((sqrt((x+a)(x+b))-x)(sqrt((x+a)(x+b))+x))/(sqrt((x+a)(x+b))+x)`

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

`=(x(a+b+ (ab)/x))/(x(sqrt(1+(a+b)/x+(ab)/x^2)+1)) = (a+b+ (ab)/x)/(sqrt(1+(a+b)/x+(ab)/x^2)+1)`

then

`lim_(x->oo)(sqrt((x+a)(x+b))-x ) = lim_(x->oo)(a+b+ (ab)/x)/(sqrt(1+(a+b)/x+(ab)/x^2)+1)=(a+b)/2`