Question

Find limits as x approaches positive and negative infinity of function f(x)= 4x^3 -x^4 ?

`4x^3-x^4`

Answer 1

`lim_(x->oo)4x^3-x^4=-oo, lim_(x->-oo)4x^3-x^4=-oo`

Explanation:

`lim_(x->oo)4x^3-x^4=4(oo^3)-oo^4=4oo-oo=oo-oo`

This is an indeterminate form and doesn't really tell us much, so let's simplify the function `4x^3-x^4.`

`lim_(x->oo)4x^3-x^4=lim_(x->oo)x^3(4-x)=oo^3(4-oo)=oo(-oo)=-oo`

We really just factored out an `x^3.`

Let's evaluate the same factored limit, but going to `-oo:`

`lim_(x->-oo)x^3(4-x=(-oo)^3(4-(-oo))=-oo(oo)=-oo`

In both cases, the function approaches `-oo.`

Note that we also could have just used rules for the end behaviors of polynomials, which tell us that when the term of highest degree has a negative coefficient and is of even degree, both ends of the function decrease (go to `-oo`).