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.

  • 0 variables
  • 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

1 more worked examplePremium

Find lim(n→∞) (1 + 2/n)^n.

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

See all 18 Limits & Continuity formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Practice

Quiz yourself on Limits & Continuity

Limit Definition of e is one of 18 Limits & Continuity formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 18 Limits & Continuity formulas