Question

How do you determine the end behavior of `f(x)=1/3x^3+5x`?

Answer 1

`-oo` to the left and `oo` to the right

Explanation:

to know the end behavior of a function you need to know two things:
`1.` if the function is positive or negative
`2.` what the base function is, and what the base function looks like

`f(x)=1/3x^3+5x`
`1.` the function is positive, so it will be increasing
`2.`the base function is `y=x^3`

`graph: y=x^3`
graph{y=x^3 [-10, 10, -5, 5]}
so knowing that `f(x)=1/3x^3+5x` is positive and base function is `y=x^3`, it will be heading towards `-oo` to the left and `oo` to the right

the `1/3` and `5x` just makes the function become a really thin `y=x^3`

`graph:y=1/3 x^3 + 5x` (if you scroll out you'll see it better)
graph{y=1/3 x^3 + 5x [-10, 10, -5, 5]}