The mathematical constant \(e\) was originally “discovered” as the limit of the following sequence:
\[e = \lim_{n \to \infty} \left(1 + \tfrac{1}{n}\right)^n.\]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 \(x = \lim_{n \to \infty} n a_n\).
\[\begin{alignat*}{2} \lim_{n \to \infty} (1+a_n)^n &= \lim_{n \to \infty} \left( 1 + \frac{x_n}{n} \right)^n &\hspace{4em}& \text{Let } x_n = n a_n \\ &= \lim_{m \to \infty} \left( 1 + \tfrac{1}{m} \right)^{mx_n} && \text{Let } m = n/x_n \\ &= \lim_{m \to \infty} \left[ \left( 1 + \tfrac{1}{m} \right)^{m} \right]^{x_n} && e^{a^b} = e^{ab}\\ &= e^x && f(a,x) = a^x \text{ is continuous}; x_n \to x \end{alignat*}\]