Question

What is the limit of `(sqrt(x^4 - 6x^2)) - x^2` as x goes to infinity?

Answer 1

`lim_(xrarroo) (sqrt(x^4 - 6x^2) - x^2) = -3`

Explanation:

`lim_(xrarroo) (sqrt(x^4 - 6x^2) - x^2)` has indeterminate form `oo-oo`.

So we'll do some algebra:

`sqrt(x^4 - 6x^2) - x^2 = ((sqrt(x^4 - 6x^2) - x^2))/1 * ((sqrt(x^4 - 6x^2) + x^2)) / ((sqrt(x^4 - 6x^2) + x^2))`

`= ((x^4-6x^2)-x^4)/(sqrt(x^4 - 6x^2) + x^2)` `" "` (has indeterminate form `oo/oo`)

`= (-6x^2)/(sqrt(x^4(1-6/x^2))+x^2)` for `x != 0`

`= (-6x^2)/(sqrt(x^4)sqrt((1-6/x^2))+x^2)` for `x != 0`

`= (-6x^2)/(x^2(sqrt(1-6/x^2)+1))` for `x != 0`

`= (-6)/(sqrt(1-6/x^2)+1)` for `x != 0`

Now as `xrarroo`, we see that the ratio `rarr (-6)/2`

So, `lim_(xrarroo) (sqrt(x^4 - 6x^2) - x^2) = -3`

Note that because the identity `sqrt(x^4)=x^2` is true for both positive and negative values of `x`, we also have

`lim_(xrarr-oo) (sqrt(x^4 - 6x^2) - x^2) = -3`