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Famous limit

An attractive way to compute this famous limit.

Problem 1 (Famous limit) Show that \[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e. \]

Answer 1 Let \(a_n = \left(1 + \frac{1}{n}\right)^n\). Then we can write \[ \ln(a_n) = n \ln\left(1 + \frac{1}{n}\right) = \frac{\ln \left(1 + \frac{1}{n}\right) - \ln 1}{\frac{1}{n}}. \] Taking the limit as \(n \to \infty\) and using the definition of the derivative, we obtain \[ \lim_{n \to \infty} \ln(a_n) = \ln^\prime(1) = 1. \] Since \(\exp: \mathbb{R} \to \mathbb{R}\) is a continuous function, exponentiating both sides gives \[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \exp(\ln(a_n)) = \exp\left(\lim_{n \to \infty} \ln(a_n)\right) = \exp(1) = e. \tag*{$\square$} \]

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