Question

How do you evaluate the limit `sqrt(x^2-2)-sqrt(x^2+1)` as x approaches `oo`?

Answer 1

`0`

Explanation:

`sqrt(x^2-2)-sqrt(x^2+1) = ((x^2-2)-(x^2+1))/(sqrt(x^2-2)+sqrt(x^2+1))`
`=(-3)/(sqrt(x^2-2)+sqrt(x^2+1)) = -(3/x^2)(1/(sqrt(1-2/x^2)+sqrt(1+1/x^2)))`

so

`lim_(x->oo)sqrt(x^2-2)-sqrt(x^2+1)=-lim_(x->oo)(3/x^2)lim_(x->oo)(1/(sqrt(1-2/x^2)+sqrt(1+1/x^2))) = 0 xx 1/2=0`