Question

What is the limit of `(2x^2+3)^(1/2) - (2x^2-5)^(1/2)` as x approaches infinity?

Answer 1

The limit is `0`

`((sqrt(2x^2+3) - sqrt(2x^2-5)))/1 ((sqrt(2x^2+3) + sqrt(2x^2-5)))/((sqrt(2x^2+3) + sqrt(2x^2-5))) = ((2x^2+3) - (2x^2-5))/(sqrt(2x^2+3) + sqrt(2x^2-5))`

`= 8/(sqrt(2x^2+3) + sqrt(2x^2-5)) = 8/(x(sqrt(2+3/x^2) + sqrt(2-5/x^2))`

As `x rarr oo`, the denominator also `rarr oo`, so lhe fraction `rarr 0`.

In anticipation:
Any proposed other method needs to also give the correct answers:

`lim_(xrarroo)(sqrt(x^2+3x) - sqrt(x^2+9x)) = -3`

`lim_(xrarr-oo)(sqrt(x^2+3x) - sqrt(x^2+9x)) = 3`