What is lim_(x->oo) (sqrt(x^2+1)-sqrt(x^2-1)) ?

1 Answer
George C.
Aug 12, 2017

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

Explanation:

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

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

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

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

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

= lim_(x->oo) 2/(2abs(x))

= lim_(x->oo) 1/abs(x)

= 0

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