CalcRef

Limits & Continuity · Definitions

Limit Definition (Epsilon-Delta)

lim⁡x→af(x)=L  ⟺  ∀ϵ>0,∃δ>0:0<∣x−a∣<δ⇒∣f(x)−L∣<ϵ\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon

The formal epsilon-delta definition of a limit. For every epsilon greater than zero, there exists a delta such that f(x) is within epsilon of L whenever x is within delta of a.

Conditions. f(x) must be defined on an open interval containing a, except possibly at a itself.

Worked examples

Prove that lim(x→3) (2x + 1) = 7 using the epsilon-delta definition.
  1. We need |f(x) - L| < ε, i.e., |(2x+1) - 7| < ε
  2. Simplify: |2x - 6| < ε → 2|x - 3| < ε → |x - 3| < ε/2
  3. Choose δ = ε/2. Then 0 < |x - 3| < δ implies |(2x+1) - 7| = 2|x-3| < 2(ε/2) = ε

Answer: δ = ε/2 satisfies the definition, confirming lim(x→3)(2x+1) = 7.

Prove that lim(x→1) x² = 1.
  1. We need |x² - 1| < ε, i.e., |x-1||x+1| < ε
  2. Restrict δ ≤ 1, so |x-1| < 1 means 0 < x < 2, thus |x+1| < 3
  3. Then |x²-1| = |x-1||x+1| < 3|x-1| < ε when |x-1| < ε/3
  4. Choose δ = min(1, ε/3)

Answer: δ = min(1, ε/3) satisfies the definition.

Related formulas

Left-Hand LimitRight-Hand Limit

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