Question

What is the limit of `(x^(6) + 4x^(4) - 8x^(2)) / [(x^(4) + 12x^(3) + 2x^(2) + 8)]^5` as x goes to infinity?

Answer 1

The limit os `0`

Explanation:

`(x^(6) + 4x^(4) - 8x^(2)) / (x^(4) + 12x^(3) + 2x^(2) + 8)^5` has, as its denominator, a degree 20 polynomial with leading coefficient `1`. Because the degree of the denominator is greater than that of the numerator, we have
`lim(xrarroo)(x^(6) + 4x^(4) - 8x^(2)) / (x^(4) + 12x^(3) + 2x^(2) + 8)^5 =0`

If we want to write some justification, we can use:

`(x^(6) + 4x^(4) - 8x^(2)) / (x^(4) + 12x^(3) + 2x^(2) + 8)^5 = (x^(6) + 4x^(4) - 8x^(2)) / (x^4(1 + 12/x + 2/x^2 + 8/x^4))^5` for `x !=0`

`= (x^20(1/x^14 + 4/x^16 - 8/x^18)) / (x^20(1 + 12/x + 2/x^2 + 8/x^4)^5)`

`= (1/x^14 + 4/x^16 - 8/x^18) / (1 + 12/x + 2/x^2 + 8/x^4)^5`

Now take limits as `xrarroo`, to get

`(0+0+0)/(1+0+0+0)^5 = 0`