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

4x^3-x^4

1 Answer
Mar 3, 2018

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).