Calculator inputs
Formula used
The formal target is: for every ε > 0, choose δ > 0 so that if 0 < |x - c| < δ, then |f(x) - L| < ε.
This calculator estimates a local bound M near the limit point and uses: δ = min(r, safety × ε / M). That makes the result conservative inside the selected neighborhood.
| Family | Model | Limit value at x → c |
|---|---|---|
| Linear | f(x) = Ax + B | L = Ac + B |
| Quadratic | f(x) = Ax² + Bx + C | L = Ac² + Bc + C |
| Cubic | f(x) = Ax³ + Bx² + Cx + D | L = Ac³ + Bc² + Cc + D |
| Absolute affine | f(x) = |Ax + B| + K | L = |Ac + B| + K |
| Reciprocal shift | f(x) = A/(x + H) + K | L = A/(c + H) + K, if c + H ≠ 0 |
| Square root shift | f(x) = A√(x + H) + K | L = A√(c + H) + K, if c + H ≥ 0 |
Because different families behave differently near special points, the verification table and graph are included to support the suggested delta.
How to use this calculator
- Select the function family that matches your problem.
- Enter the coefficients and the limit point c.
- Set epsilon, the neighborhood radius, and the safety factor.
- Press Calculate Delta to generate L, M, and the suggested delta.
- Read the verification status, inspect the graph, and review the nearby data rows.
- Download the generated table as CSV or export the summary as PDF.
Example data table
Example for the linear model f(x) = 3x - 1 at x → 2, where L = 5.
| Epsilon ε | Bound M | Suggested delta δ | Interpretation |
|---|---|---|---|
| 0.90 | 3 | 0.30 | Any x within 0.30 of 2 keeps f(x) within 0.90 of 5. |
| 0.60 | 3 | 0.20 | A smaller epsilon requires a tighter x-neighborhood. |
| 0.30 | 3 | 0.10 | The allowed delta shrinks as the tolerance shrinks. |
Frequently asked questions
1) What does epsilon represent here?
Epsilon is the allowed vertical error around the limit value. It measures how close f(x) must stay to L once x is chosen near the limit point.
2) What does delta mean in the result?
Delta is the horizontal distance from the limit point. If x stays within that distance, the function values should remain inside the epsilon band shown in the result.
3) Why is a safety factor included?
The safety factor makes delta more conservative. A value below 1 slightly shrinks the result, which helps when you want cleaner verification against sampled nearby points.
4) Is this a formal proof generator?
No. It is a structured calculator and numerical checker. It helps you estimate a good delta and inspect behavior, but textbook proofs may still require symbolic derivations.
5) Why can the reciprocal model fail?
The reciprocal family becomes undefined when x + H equals zero. If the limit point lands on that singularity, the function has no finite value there, so the calculator stops.
6) Why can the square root model fail?
Square root models need x + H to stay nonnegative. If the limit point makes that expression negative, the function is outside its real-number domain and cannot be evaluated.
7) What does the estimated local bound M do?
M measures how quickly the function changes near the limit point. Larger values mean the graph changes faster, so the calculator usually returns a smaller delta for the same epsilon.
8) When should I increase the search radius?
Increase the radius when you want a broader local scan. Reduce it when the function changes sharply or approaches domain restrictions, since smaller neighborhoods often produce safer delta estimates.