# Can someone explain why the limit of (1 + 4/n)^n as n approaches infinity = e^4?

### 1 Answer

Relevance

- 6 years agoFavourite 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

Still have questions? Get answers by asking now.