Urgent limit question thanks :)

lim x -> ∞ (x)(e^(-x)

thanks for your help :)

lim (x)(e^(-x)
= lim 1 / [e^(x) / x]
= lim 1 / [ 1 + x + x^2/2! + x^3/3! + ....] /x
= lim 1 / [1/x +1 + x/2! + x^2/3! +.....]
= lim x -> ∞ 1 / ∞
= 0
answer

x -> ∞ : (x)*e^(-x)
let's rewrite this problem:
x -> ∞ : x / (e^x)
now we'll apply L'hopital's rule (basically the limit of (a/b) is equal to the limit of (derivitive of a)/(derivitive of b))
x -> ∞ : x' / (e^x)'
x -> ∞ : 1 / (e^x)

taking the limit of this expression yeilds infinity, so
x -> ∞ : (x)*e^(-x) = ∞