Determine if the limit?

Question

lim x^4e^3x
x->-infinity
exists and if it does, find its value
how do you do. plz explain the easiest way. thnx

Answer 1

A limit does exist, and it is zero. For each unit integer decrease in x, x<0, the e-term is reduced by a factor of e-3. For the same decrease, the x-term is INCREASED by
1+ [4n^3+6n^2+4n+1]/n^4.
This factor increases at a slower rate than does the decrease of the e-term, so their product will be less than 1.

Answer 2

You can write your lim as:
lim(x^4/e^3x) for x -> + inf.
In this case limes exists as inf/inf and can be found by successive L'hopital rule. Therefore:

lim(x^4/e^3x) = lim((4x^3)/(3e^3x)) = lim((3*4*x^2)/(3*3*e^3x))
= lim((2*3*4*x)/(3*3*3*e^3x)) = lim((2*3*4)/(3*3*3*3*e^3x)) = 0.