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What is the limit of #sqrt(x^2+8x-5)-sqrt(x^2-6x+2)# as x approaches infinity?

Answer 1

#7#

#(sqrt(x^2+8x-5)-sqrt(x^2-6x+2))(sqrt(x^2+8x-5)+sqrt(x^2-6x+2))/(sqrt(x^2+8x-5)+sqrt(x^2-6x+2)) =# #=(x^2+8x-5-(x^2-6x+2))/(sqrt(x^2+8x-5)+sqrt(x^2-6x+2))=# #=(14x-7)/(sqrt(x^2+8x-5)+sqrt(x^2-6x+2)) =# #=(14x-7)/(x(sqrt(1+8/x-5/x^2)+sqrt(1-6/x+2/x^2))) =# #(14-7/x)/(sqrt(1+8/x-5/x^2)+sqrt(1-6/x+2/x^2))#

Finally

#lim_{x->oo}(sqrt(x^2+8x-5)-sqrt(x^2-6x+2))=14/2=7#

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Answer 2

Axel Alvarez

The limit of sqrt(x^2+8x-5)-sqrt(x^2-6x+2) as x approaches infinity is 7.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.