Question

How do you find the limit of `(2x^3+3x^2cosx)/(x+2)^3` as `x->oo`?

Answer 1

The limit has indeterminate initial form `oo/oo`. We'll try to make the denominator go to someting oher than `oo`.

Explanation:

We want the limit as `xrarroo`, we aren't worried about what happens at `0`.

For `x != 0`,

`(2x^3+3x^2cosx)/(x+2)^3 = (x^3(2+(3cosx)/x))/(x(1+2/x))^3`

`= (x^3(2+(3cosx)/x))/(x^3(1+2/x)^3)`

`= (2+(3cosx)/x)/(1+2/x)^3`

As `xrarroo`, the denominator goes to `1` and the numerator goes to `2`

So the limit of the ratio as `xrarroo` is `2`