# Epsilon-delta definition

The limit of f(x) at a is L if, for every tolerance epsilon > 0, there is a distance delta > 0 so that x within delta of a gives f(x) within epsilon of L.

Maths (university) · Part 02: Limits and continuity · Card 9 of 9 · Last updated 2026-10-19

## Analogy

A dart game: however small the bullseye (epsilon) is, you can find a steady-enough aim (delta) to hit it.

## Example

**Show lim (2x + 1) = 7 at x = 3**

|f(x) - 7| = |2x - 6| = 2|x - 3|.
We want this less than epsilon.
Choose delta = epsilon / 2.
Then |x - 3| < delta gives |f(x) - 7| < epsilon.

## In practice

It is the exact definition behind all limit rules, and the starting point of formal analysis.

## Practice

### Flashcards

- **In epsilon-delta for 2x + 1 near 3, which delta works?** delta = epsilon / 2.

## Questions people ask

### How do I choose delta in an epsilon-delta proof?

Start from |f(x) - L| and rewrite it in terms of |x - a|. Then find a delta that forces it below epsilon. For 2x + 1 near 3 the difference is 2|x - 3|, so delta = epsilon over 2 works. Check by substituting back.

### Why do we need such a formal definition of limit?

Words like 'gets close' are vague and can fail on tricky functions. The epsilon-delta definition is exact, so proofs can be checked line by line. It also lets mathematicians prove the limit laws and the squeeze theorem rather than just trusting them.

## Related concepts

- [Limit](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-1)
- [One-sided limit](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-2)
- [Limit laws](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-3)
- [Squeeze theorem](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-4)

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