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

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    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

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