Epsilon Delta Limit Proof Calculator

Calculator Inputs

Function model

Epsilon

Approach value a

Claimed limit L

Local radius r

Safety factor

Sample check count

Delta unit label

Linear m

Linear b

Quadratic A

Quadratic B

Quadratic C

Polynomial coefficients

Use constant first. Example: 1,-2,1 means 1 - 2x + x².

Rational numerator coefficients

Rational denominator coefficients

Square-root c

Absolute value c

Formula Used

The main definition is: for every epsilon greater than zero, there is a delta greater than zero such that 0 < |x - a| < delta implies |f(x) - L| < epsilon.

This calculator uses a local bound. If |f(x) - f(a)| <= M|x - a|, then a safe choice is delta <= (epsilon - |f(a) - L|) / M.

For removable forms, the expression is simplified before applying the proof. For square-root forms near zero, the calculator can use delta <= epsilon squared.

How to Use This Calculator

Select a function model from the list.
Enter epsilon, the approach value, and the claimed limit.
Leave the claimed limit blank to use the computed limit.
Set a local radius for the proof interval.
Use a safety factor below one for a stricter delta.
Press Calculate Proof to see the result above the form.
Download the CSV or PDF file for records.

Example Data Table

Model	Function	a	Epsilon	Computed Limit	Typical Delta
Linear	3x + 1	2	0.06	7	0.02
Quadratic	x² - 2x + 1	2	0.05	1	Depends on radius
Removable	(x² - a²) / (x - a)	3	0.01	6	0.01
Absolute	\|x - 4\|	1	0.20	3	0.20

Understanding Epsilon Delta Proofs

An epsilon delta proof turns a limit idea into a testable statement. The goal is simple. For every positive error band epsilon, find a positive input band delta. When x stays inside delta of a, the function value must stay inside epsilon of L. This calculator helps build that chain with clear bounds.

Why Bounds Matter

Most proofs fail because the bound is not controlled. A useful proof does not only test points. It shows why every nearby point works. Linear functions use a slope bound. Quadratics use a local factor around the chosen point. Polynomial and rational options use derivative style bounds. These bounds support a formal inequality path.

Construction Context

Construction work often uses tolerances, margins, and acceptance limits. Epsilon delta logic is similar. Epsilon is the allowed output tolerance. Delta is the allowed input tolerance. A drawing, cut length, slope, or material measurement may need controlled variation. The calculator can show how a small input change protects the final value.

Proof Workflow

Start with a function model. Enter the approach value a. Enter the claimed limit if you want to test it. Leave it blank to use the computed limit. Choose epsilon and a local radius. A smaller radius usually gives a stronger, safer proof. The safety factor shrinks the final delta, which adds a practical margin.

Interpreting Results

The result gives the computed limit, claim error, bound constant, raw delta, and final delta. If the claim does not match the computed limit, the proof may fail for small epsilon values. The warning explains this. Sample points are included only as checks. They do not replace the proof.

Exporting Work

Use the CSV option for spreadsheet review. Use the document export for a compact proof record. The exported data includes the main inputs, delta, formulas, and sample checks. Keep the local radius and assumptions with the proof, because they explain where the bound is valid.

Good Input Habits

Use realistic numbers and units. Avoid zero epsilon values. Check the denominator warning for rational forms. Increase sample counts when reviewing tight tolerances. Compare the final delta with field precision. If the needed delta is too small, improve measurement methods or relax the output requirement.

FAQs

What does epsilon mean?

Epsilon is the allowed output error. It defines how close f(x) must be to the claimed limit L. Smaller epsilon values require tighter input control.

What does delta mean?

Delta is the allowed input distance from the approach value a. If x stays inside delta, the proof should keep f(x) inside the epsilon band.

Can this prove every possible limit?

No. It supports common structured models. General symbolic proofs may need manual algebra, domain analysis, or theorem support beyond this calculator.

Why does the claimed limit matter?

The proof target is L. If L differs from the computed limit, the statement usually fails for small epsilon values. The calculator warns about that mismatch.

Why use a local radius?

The local radius limits the interval where bounds are built. Smaller intervals often produce better constants and safer delta values.

Are sample checks enough for a proof?

No. Sample checks are only numerical evidence. The proof depends on the inequality bound that controls all nearby x values.

What does the safety factor do?

The safety factor reduces the raw delta. This adds margin and helps avoid borderline values in practical tolerance work.

Why can rational functions show warnings?

A rational function can behave poorly near denominator zeros. If the denominator gets very small, reduce the local radius or review the proof manually.