Section 2.7

The Epsilon-Delta Definition

The rigorous foundation that legitimized calculus and banished Berkeley's "ghosts of departed quantities."

The Formal Definition

The Definition

means:

For every (however small), there exists a such that

This definition replaces "getting closer" with a static logical statement. It avoids the concept of "motion" entirely, treating the limit as a condition of the function's mapping.

Pedagogical Strategy: The Game

The best way to understand this is as a game between a Skeptic (Student) and a Prover (Mathematician).

Skeptic

"I bet you can't get the output within of the limit."

Prover

"I can. I just restrict the input to within of ."

Skeptic

"Okay, try !"

Prover

"Then I restrict input to ."

Victory Condition

If the Prover has a formula to find a for any possible , the limit is proven.

Detailed Example: Linear Proof

Goal: Prove

Step 1: Analyze

We want

Step 2: Simplify

Step 3: Choose δ

Set

Step 4: Proof

If , multiplying by 2 gives , which simplifies to . ∎

Interactive Visualization

Epsilon-Delta Visualizer

For f(x) = 2x + 1 at x → 3, L = 7. Adjust ε and δ to see how they relate.

ε (epsilon) - error tolerance on outputε = 1.00

δ (delta) - input restrictionδ = 0.50

✓ Valid!Required: δ ≤ ε/2 = 0.50

When |x - 3| < 0.50, then |f(x) - 7| < 1.00 ✓

ε-band: |f(x) - L| < ε (output tolerance)

δ-band: |x - a| < δ (input restriction)

Key insight: For this linear function, choosing δ = ε/2 always works. The Prover's formula guarantees success for ANY ε the Skeptic chooses!