Calculus I · Limits and Continuity · lesson
Epsilon-Delta Definition: An Introduction
Learn Epsilon-Delta Definition: An Introduction with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style p
Where this chapter fits
Chapter 6: Formal limits
Translate the intuitive neighborhood picture into epsilon-delta language, constructive proofs, graph windows, and counterexamples.
Reading lens: How small must the input window be to force every allowed output into the requested tolerance band? Keep that question in view while reading Epsilon-Delta Definition: An Introduction; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Bisection Method After the IVT to Epsilon-Delta Proofs for Linear Functions. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Translate "arbitrarily close" into inequalities; understand the roles of and ; verify simple formal limit proofs.
Making Limits Precise
Why the Informal Definition Needs Numbers
The informal definition says that can be made as close as desired to by taking sufficiently close to . The phrase "as close as desired" must be made numerical.
Suppose a customer says, "Make the board close to 10 feet long." A builder needs a tolerance: within one inch? one millimeter? The symbol names the allowed output error. The symbol tells how close the input must be to guarantee that error.
The distance between and is
The distance between and is
Formal \(\eps\)-\(\delta\) Definition
We say
if
In words:
For every positive output tolerance , there is a positive input tolerance such that every allowed input within of , except itself, produces an output within of .
| Expression | Meaning |
|---|---|
| The input lies in the horizontal interval . | |
| The output lies in the vertical band . | |
| The target input itself is excluded. | |
| The argument must work for every requested accuracy. | |
| We may choose an input tolerance depending on . |
Source & rights
Original instruction with traceable references.
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