Determine the limit

Question

L’Hopital’s Rule: Determine the Limit
a. lim x→∞
x^3e^-x

Answer 1

This lim = 0/0

x^3e^-x

make into fraction

x^3/e^x

take derivative of top and bottom

3x^2/e^x this will be 0/0

so do it again

6x/e^x lim = 0/0

6/e^x lim = 0

 

If i am reading this wrong and it is x^{3e^-x} then lim = ∞⁰

So say

y = x^{3e^-x} then ln(y) = ln(x^{3e^-x})

ln(y) = 3e^-x*ln(x) then turn it into a fraction

ln(y) = ln(x)/3e^x lim = ∞/∞

l'hop

ln(y) = (1/x)/3e^x = 1/x3e^x lim = 1//∞*∞ = 0

y=1