How ca I find the limit of 1/[sqrt(x^2-x+1)+sqrt(x^2-x-1)] as x goes to infinity?

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Answer 1 · 2018-01-31T23:01:48

#0#

#1/(sqrt(x^2-x+1)+sqrt(x^2-x-1))#

Multiply by #(sqrt(x^2-x+1)-sqrt(x^2-x-1))# ( conjugate )

#(sqrt(x^2-x+1)-sqrt(x^2-x-1))/((sqrt(x^2-x+1)+sqrt(x^2-x-1))(sqrt(x^2-x+1)-sqrt(x^2-x-1)))#

#(sqrt(x^2-x+1)-sqrt(x^2-x-1))/2#

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

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

#=0/2=0#

#:.#

#lim_(x->oo)1/(sqrt(x^2-x+1)+sqrt(x^2-x-1))=0#

Note

Use of the conjugate.

#(a+b)(a-b)=a^2-b^2#