Epsilon-Delta Definition of a Limit – AP Calculus AB | ShowMeClass

εδ

ε-δ Definition of a Limit

AP Calculus AB · Unit 1: Limits & Continuity

Learning Objectives

State the formal ε-δ definition of a limit

Visualize how δ constrains x to keep f(x) within ε of L

Adjust ε and find a suitable δ

Understand why ε-δ is the rigorous foundation of limits

Formal Definition

For every ε > 0, there exists a δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε.

Visual Guide

ε-band: L−ε to L+ε (horizontal)

δ-band: a−δ to a+δ (vertical)

f(x) curve

Point (a, L)

Intuition

Think of ε as a challenge: "I want f(x) within ε of L." Your job is to find a response δ: "If x is within δ of a, f(x) stays in the ε-band." If you can always find such a δ no matter how small ε gets, the limit exists.

ε-δLimitsAP Calc ABUnit 1

a =—

L =—

ε =—

δ =—

Status—

ε (epsilon) 1.00

δ (delta) 0.50

Point a 2.00

Function f(x)

ε-δ Analysis

📝 Quick Quiz▼

1. In the ε-δ definition, ε represents:

How close x must be to a How close f(x) must be to L The value of the limit The slope at a

2. For f(x) = 2x, lim(x→3) = 6. If ε = 0.1, the largest δ is:

0.1 0.05 0.2 0.01

3. If no δ works for a given ε, this means:

The limit is infinite ε is too large The limit does not equal L δ must be negative