CalcRef

Limits & Continuity · Special Limits

Limit Definition of e

lim⁡n→∞(1+1n)n=e\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

The number e (≈ 2.71828) defined as a limit. Equivalently, lim(x→0) (1+x)^(1/x) = e.

Worked examples

Find lim(x→0) (1+3x)^(1/x).
  1. Rewrite as [(1+3x)^(1/(3x))]³
  2. Let u = 3x. As x→0, u→0.
  3. lim(u→0) (1+u)^(1/u) = e
  4. So the limit = e³

Answer: e³ ≈ 20.086

Find lim(n→∞) (1 + 2/n)^n.
  1. Rewrite as [(1 + 2/n)^(n/2)]²
  2. Let m = n/2. As n→∞, m→∞.
  3. [(1+1/m)^m]² → e²

Answer: e² ≈ 7.389

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