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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.

  • 2 worked examples
  • 18 in Limits & Continuity

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

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Find lim(n→∞) (1 + 2/n)^n.

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Common questions

What is the Limit Definition of e?

The number e (≈ 2.71828) defined as a limit. Equivalently, lim(x→0) (1+x)^(1/x) = e. It is one of the limits & continuity formulas in the CalcRef reference.

How do you use the Limit Definition of e?

Worked example. Find lim(x→0) (1+3x)^(1/x). Rewrite as [(1+3x)^(1/(3x))]³. Let u = 3x. As x→0, u→0. lim(u→0) (1+u)^(1/u) = e. So the limit = e³. Answer: e³ ≈ 20.086.

Practice

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