Lim x → ∞{(x+a)(x+b)}^1/2 -x) equals?

1 Answer
Cesareo R.
Aug 2, 2017

(a+b)/2a+b2

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) √(x+a)(x+b)−x=(√(x+a)(x+b)−x)(√(x+a)(x+b)+x)√(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)(x+b)−x=x2+(a+b)x+ab−x2√(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)=x(a+b+abx)x(√1+a+bx+abx2+1)=a+b+abx√1+a+bx+abx2+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

Related questions
Impact of this question
1854 views around the world
You can reuse this answer
Creative Commons License