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Can someone explain why the limit of (1 + 4/n)^n as n approaches infinity = e^4?
1 Answer
- 6 years agoFavorite Answer
We can use L'hopital's rule to show it, but (1 + r/n)^n going to e^r as n goes to infinity is just one of those definitions of e
L = (1 + r/n)^n
ln(L) = n * ln(1 + r/n)
ln(L) = ln(1 + r/n) / (1/n)
u = 1/n
n goes to infinity, u goes to 0
ln(L) = ln(1 + r * u) / u
As u goes to 0, we get ln(1) / 0 => 0/0, which is indeterminate
f(u) = ln(1 + r * u)
f'(u) = r / (1 + r * u)
g(u) = u
g'(u) = 1
(r / (1 + r * u)) / 1 =>
r / (1 + r * u)
u goes to 0
r / (1 + 0) =>
r
ln(L) = r
L = e^r
So we see that the limit of (1 + r/n)^n as n goes to infinity is e^r
