How do you find the limit of [(x^2+x)^(1/2)-x] as x approaches infinity?

1 Answer
May 10, 2016

lim_(x->oo)(sqrt(x^2+x)-x)=1/2

Explanation:

lim_(x->oo)(sqrt(x^2+x)-x) = lim_(x->oo)((sqrt(x^2+x)-x)(sqrt(x^2+x)+x))/(sqrt(x^2+x)+x)

=lim_(x->oo)(x^2+x-x^2)/(sqrt(x^2+x)+x)

=lim_(x->oo)x/(sqrt(x^2+x)+x)

=lim_(x->oo)1/(sqrt(x^2+x)/x+x/x)

=lim_(x->oo)1/(sqrt((x^2+x)/x^2)+1)

=lim_(x->oo)1/(sqrt(1+1/x)+1)

=1/(sqrt(1+0)+1)

=1/2