How do you find the limit of #(x-sqrt(x^2-x))# as x approaches infinity?
1 Answer
Jun 24, 2016
#lim_(x->oo) (x - sqrt(x^2-x)) = 1/2#
Explanation:
#lim_(x->oo) (x - sqrt(x^2-x))#
#=lim_(x->oo) (1/2+(x-1/2) - sqrt((x-1/2)^2-1/4))#
#=1/2 + lim_(t->oo) (t-sqrt(t^2-1/4))#
#=1/2 + lim_(t->oo) ((t-sqrt(t^2-1/4))(t+sqrt(t^2-1/4)))/(t+sqrt(t^2-1/4))#
#=1/2 + lim_(t->oo) (t^2-(t^2-1/4))/(t+sqrt(t^2-1/4))#
#=1/2 + lim_(t->oo) 1/(4(t+sqrt(t^2-1/4)))#
#=1/2#
