Question regarding limits of sequences and functions.

Question

Given that lim n→∞(1+1/n)^n = e :

(a) Show that lim n→∞(1-1/n)^n = lim k→∞(k/k+1)^(k+1)

(b) Hence, show that lim n→∞(1-1/n)^n = 1/e

(c) Therefore, using the Squeeze Theorem, find lim n→∞(n!/n^n).

Answer 1

I can help you with 2 parts; I don't see how to use the "Squeeze" Theorem in the 3rd part.

1-(1/n)=(n-1/n)

Substitute n=k+1 and

[1-(1/n)]ⁿ becomes [k/(k+1)]^k+1 = {1/[1+(1/k)]}^k+1 = [1+(1/k)]^-k-1

I think this last expression = e^-1'

lim n->∞ n!/nⁿ = 0 at least one of the factors in the quotient (1/n) goes to 0 as n goes to ∞ and at worst all

the other terms approach 1. I'm not sure how the first 2 parts are related to the 3rd, nor do I see how to use the so-called "Squeeze" Theorem in this case.

I"m sorry I cannot be more helpful.