Calculus I › Limits › full formula sheet
The definition of e
The number that compound interest converges to — no matter how often you compound.
On this page e ≈ 2.71828… is defined as this limit (it is irrational, in fact transcendental). Equivalent forms: lim(n→∞)(1+1/n)n = e and lim(x→0)(1+x)1/x = e.
Before this lesson: Limits at infinity
Notation in this lesson
- e
- ≈ 2.71828 — the base of natural growth
- limn→∞
- the limit as n grows without bound
- (1+1/n)n
- the compounding expression: more frequent compounding, smaller pieces
Where it comes from
Put $1 in a bank at 100% annual interest. Compounded yearly: (1+1)1 = $2. Compounded monthly: (1+1/12)12 ≈ $2.61. Compounded daily: (1+1/365)365 ≈ $2.71. Compounded every second — do you get rich?
The naive guesses:
The intuition: (1+1/n)n balances two forces — the base (1+1/n) shrinking toward 1, the exponent n growing toward infinity. It’s a 1∞ indeterminate form, and the balance point is e. The derivation proves the balance point exists; the digits come from computing.
Derivation
We prove the limit exists (then we may name it e) via the binomial theorem + the Monotone Convergence Theorem: increasing and bounded above ⇒ convergent.
Key steps shown; the existence argument is complete. Note what this proves: existence, not digits — the 2.71828… comes from computing partial sums (or (1+1/1000)1000 ≈ 2.717). Analysis guarantees the limit is there; arithmetic tells you its address. (e is irrational — in fact transcendental — but that’s another page.)
How to use it
The procedure, every time:
- Spot the shape: (1 + □)1/□ with □ → 0, or (1+1/□)□ with □ → infinity — a 1∞ form.
- Rewrite to match exactly: the fraction’s denominator and the exponent must be the same □. (1+2/x)x = [(1+2/x)x/2]² → e².
- Apply: the matched core → e; outer powers/constants follow by the power law.
- Variants: lim(n→∞)(1+1/n)n, lim(x→∞)(1+1/x)x, lim(x→0)(1+x)1/x are all e (substitute n = 1/x for the last).
The exponent must match
(1+1/n)n² = [(1+1/n)n]n ≈ en → infinity — the exponent outruns the base. (1+1/n²)n → 1 — the base approaches 1 too fast (n·ln(1+1/n²) ≈ 1/n → 0). Only the matched form gives e. Mismatches are a favorite exam trap.
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the definition: lim(n→∞) (1+1/n)n
- Recognize the shape: exactly (1+1/□)□ with □ = n → infinity.
- Apply: this is e, by definition. Answer: e.
Your turn: limn→∞ (1+1/n)2n = ?
Answer: e².
Write it as [(1+1/n)n]². The inside → e, so the square → e² by the power law.
Example 2 — scaled: lim(x→∞) (1+2/x)x
- Match the form: the fraction says 2/x but the exponent says x. Rewrite: (1+2/x)x = [(1+2/x)x/2]².
- Inner limit: as x → infinity, x/2 → infinity, so (1+2/x)x/2 → e (with □ = x/2).
- Outer power: e² by the power law. Answer: e².
Your turn: limx→∞ (1+3/x)x = ?
Answer: e³.
Rewrite as [(1+3/x)x/3]³: the inside → e (with □ = x/3 → ∞), then cube → e³.
Example 3 — fractional power: lim(x→∞) (1+1/(2x))x
- Match: (1+1/(2x))x = [(1+1/(2x))2x]1/2.
- Inner: (1+1/(2x))2x → e (with □ = 2x).
- Outer: e1/2 = √e.
Your turn: limx→∞ (1+1/(3x))x = ?
Answer: ∛e = e1/3.
Write as [(1+1/(3x))3x]1/3 → e1/3.
Example 4 — the x → 0 variant: lim(x→0) (1+x)1/x
- Substitute n = 1/x: as x → 0, n → ±infinity, and (1+x)1/x = (1+1/n)n.
- Apply: → e (both one-sided n-limits give e).
Your turn: limx→0 (1+2x)1/x = ?
Answer: e².
Rewrite as [(1+2x)1/(2x)]² → e².
Memorization tips
- The compound-interest story: (1+1/n)n is $1 at 100% compounded n times. n = 12 gives 2.61, n = 365 gives 2.71, n → infinity gives e. The story is the definition.
- The exponent must match the denominator: (1+1/□)□ → e. Mismatches explode (² → infinity) or collapse (→ 1).
- Generalize: lim(x→∞)(1+k/x)x = ek. The k just rides along into the exponent.
- 1∞ is indeterminate: never “equals 1”. This page’s e, the mismatches’ infinity and 1 — three answers from one shape.
- Existence vs digits: the binomial/monotone proof shows the limit exists; computing (1+1/1000)1000 ≈ 2.717 gives the digits.
- Why e matters: it’s the unique base with d/dx[ex] = ex — the fixed point of differentiation. This limit is where that base comes from.
Final challenge
Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and this law is yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is the definition of e?
e = lim(n→∞)(1+1/n)ⁿ ≈ 2.71828… — the number compound interest converges to. Equivalent forms: lim(x→∞)(1+1/x)ˣ = e and lim(x→0)(1+x)^{1/x} = e.
Why doesn't infinite compounding give infinite money?
Each extra compounding period adds less than the last: the sequence (1+1/n)ⁿ is increasing but bounded above by 3, so it converges to e ≈ 2.71828 instead of exploding.
How do you prove the limit exists?
Binomial expansion shows every term grows with n (increasing sequence) and each term ≤ 1/k!, with Σ1/k! < 3 (bounded above). Increasing + bounded ⇒ convergent, by the Monotone Convergence Theorem.
What is lim(x→∞)(1+2/x)ˣ?
e². Rewrite as [(1+2/x)^{x/2}]²: the inner matched form → e, and the outer square gives e². In general, lim(1+k/x)ˣ = eᵏ.
What goes wrong with (1+1/n)^{n²}?
The exponent must match the denominator. (1+1/n)^{n²} = [(1+1/n)ⁿ]ⁿ ≈ eⁿ → ∞ — the exponent outruns the base. Only the matched form gives e.
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