Exponential limit

The mathematical constant $e$ was originally “discovered” as the limit of the following sequence:

e = lim_{n \to \infty} {(1 + \frac{1}{n})}^{n} .

The following theorem presents a useful generalization of this result.

Theorem: Suppose the sequence $n a_{n}$ has a limit as $n \to \infty$ . Then $lim_{n \to \infty} (1 + a_{n})^{n} = \exp (lim_{n \to \infty} n a_{n}) .$

Proof: Let $x = lim_{n \to \infty} n a_{n}$ .

\begin{aligned} lim_{n \to \infty} (1 + a_{n})^{n} & = lim_{n \to \infty} {(1 + \frac{x_{n}}{n})}^{n} & Let x_{n} = n a_{n} \\ = lim_{m \to \infty} {(1 + \frac{1}{m})}^{m x_{n}} & Let m = n / x_{n} \\ = lim_{m \to \infty} {[{(1 + \frac{1}{m})}^{m}]}^{x_{n}} & e^{a^{b}} = e^{a b} \\ = e^{x} & f (a, x) = a^{x} is continuous; x_{n} \to x \end{aligned}