---
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: 2026-11-12
---

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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

- Interpret the tolerance, input bound and quantifier order in an epsilon-delta statement
- Calculate and check input-output error bounds for linear and quadratic functions, including counterexamples to proposed bounds
- Explain why a limit proof must cover arbitrarily small positive tolerances

## Lesson: Promises with epsilon and delta

A finite limit claim fixes an input point a and target L. The domain must contain inputs other than a arbitrarily close to a. Epsilon is a positive tolerance for the output error |f(x)-L|. A challenger may choose any epsilon. You then choose a positive input distance delta that keeps every permitted output within epsilon of L.

Assume domain inputs other than a occur arbitrarily close to a. The limit is L if, for every epsilon > 0, some delta > 0 makes every domain input x with 0 < |x-a| < delta satisfy |f(x)-L| < epsilon. Delta may depend on epsilon, a and the function, but not on the individual input x. One delta must work for all permitted inputs.

For f(x) = mx+b with m > 0 and L = ma+b, the output error is m|x-a|. An input distance below delta gives output error below m delta. Thus delta = epsilon/m works, and every smaller positive delta works too. Given delta, m delta is the boundary output-error bound.

The condition 0 < |x-a| < delta means a-delta < x < a+delta, with x = a excluded. The endpoints are excluded too because the inequalities are strict. The value f(a) can be missing or different from L without changing a limit. For example, 0 < |x-6| < 0.4 excludes 6, 5.6 and 6.4.

For a line with nonzero slope m, use its absolute value: |f(x)-L| = |m||x-a|, so delta = epsilon/|m| works. Smaller positive distances work too. A negative slope still gives a positive error. If m = 0, f is constant and L = b. The output error is zero, so every positive delta works; do not divide by zero.

Checking one tolerance does not prove a limit. A smaller epsilon may need a smaller delta. However, a delta that works for one epsilon also works for every larger epsilon, since the output requirement becomes looser. To prove a limit, give a working delta for an arbitrary positive epsilon.

For f(x) = x² near 0 with L = 0, the output error is x². The inequality x² < epsilon is equivalent to |x| < sqrt(epsilon), so delta = sqrt(epsilon) works. It is the largest working delta. Any larger delta includes x = sqrt(epsilon), whose output error equals epsilon and fails the strict requirement.

**Example.** Prove that 5x-2 tends to 3 at x = 1. Let epsilon > 0 and choose delta = epsilon/5. Assume 0 < |x-1| < delta. Then |(5x-2)-3| = 5|x-1| < 5 delta = epsilon. This proves the claim for every positive epsilon. The proof uses a strict inequality, not just equality at the boundary.

*(drawing: The assumed input bound gives the required strict output bound.)*

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

*Original epsilon-delta lesson, questions and flashcards written for this learning dataset.*

## 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)

## Optional connections

- [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)