Question

How do you find the limit in the following question?

`limxrarr∞`

`(x^3-2x^2)^(1/3) + (x^3-x^2)^(1/3) - (8x^3+4x^2)^(1/3)`

Answer 1

`-4/3`

Explanation:

We use the binomial expansion for general index `n` :

`(1+z)^n = 1+nz+(n(n-1))/(2!)z^2+...`

for each of the three terms

`(x^3-2x^2)^(1/3) = x(1-2/x)^(1/3)`
`qquad = x(1-1/3 times 2/x+O(1/x^2))`
`qquad = x-2/3+O(1/x)`

`(x^3-x^2)^(1/3) = x(1-1/x)^(1/3)`
`qquad = x(1-1/3 times 1/x+O(1/x^2))`
`qquad = x-1/3+O(1/x)`

`(8x^3+4x^2)^(1/3) = 2x(1+1/(2x))^(1/3)`
`qquad = 2x(1+1/3 times 1/(2x)+O(1/x^2))`
`qquad = 2x+1/3+O(1/x)`

Combining these, we get

`(x^3-2x^2)^(1/3) + (x^3-x^2)^(1/3) - (8x^3+4x^2)^(1/3)`
`qquad = [x-2/3+O(1/x)]+[x-1/3+O(1/x)]`
`qquad -[2x+1/3+O(1/x)]`
`qquad = -4/3+O(1/x)`

Thus, the limit of this expression as `x to oo` is `-4/3`