---
title: "The Epsilon-Delta Definition of a Limit"
description: "State what a limit claims with no appeal to motion: for every tolerance around the limiting value there is a distance around the point that keeps outputs inside it. Prove a simple limit by producing t"
canonical: https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk
source: https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk.md
retrieved: 2026-09-12
---

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# The Epsilon-Delta Definition of a Limit

State what a limit claims with no appeal to motion: for every tolerance around the limiting value there is a distance around the point that keeps outputs inside it. Prove a simple limit by producing that distance.

Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk

## Ready when they can

- Read an epsilon-delta statement as a challenge and a response
- Produce a delta that works for a given epsilon on a linear function
- Explain why the definition has to hold for every epsilon, not one small one

## Lesson: Promises with epsilon and delta

You claim that f(x) tends to L as x tends to a. Your challenger picks any positive tolerance epsilon around L. You must answer with a positive distance delta around a so that every x within delta of a, except a itself, lands with f(x) within epsilon of L.

**Example.** Take f(x) = 3x + 1 near x = 2, where the limit is 7. The output error is |3x + 1 - 7| = 3|x - 2|. For any epsilon, delta = epsilon / 3 works, since 3|x - 2| then stays below epsilon.

One small epsilon is never enough, since the challenger may demand any closeness at all. The definition says for every epsilon, so your answer must cover each demand. Note that 0 < |x - a| leaves out x = a, since limits track nearness, not the value at the point.

**Tip.** For a line f(x) = mx + b, divide by the slope: delta = epsilon / |m|. Check by putting it back: |f(x) - L| becomes |m| times |x - a|, which lands below epsilon. For y = x squared near 0, x squared < epsilon means |x| < sqrt(epsilon), so delta = sqrt(epsilon).

**Recap.** Epsilon is the demanded closeness, delta is your answer, and the pair must work for every epsilon.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Continuity and the Intermediate Value Theorem](https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV)
- [Proof by Contradiction and Counterexample](https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8)

## Opens up

- [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv)
- [Convergence of Sequences, Rigorously](https://lightmysky.com/learn/mathematics/convergence-of-sequences-rigorously-mt_yB5M-DCH8V)