Calculator Inputs
Approach Path Summary
| Path | Points Used | Last Approximation | Internal Range | Status |
|---|---|---|---|---|
| y = x | 6 | 0 | 0 | Sample preview |
| y = 0 | 6 | 1 | 0 | Sample preview |
Plotly Graph
The chart compares sampled function values against distance from the target point for several approach paths.
Example Data Table
This example uses the function (x² − y²) / (x² + y²) near (0, 0). It shows why path testing matters.
| Path | Step | x | y | Distance | f(x, y) |
|---|---|---|---|---|---|
| y = x | 1 | 1 | 1 | 1.414214 | 0 |
| y = x | 0.5 | 0.5 | 0.5 | 0.707107 | 0 |
| y = x | 0.25 | 0.25 | 0.25 | 0.353553 | 0 |
| y = 0 | 1 | 1 | 0 | 1 | 1 |
| y = 0 | 0.5 | 0.5 | 0 | 0.5 | 1 |
| y = 0 | 0.25 | 0.25 | 0 | 0.25 | 1 |
Formula Used
General limit idea: A multivariable limit exists when f(x, y) approaches the same value from every nearby path.
Numerical approach:
1. Choose a target point (a, b).
2. Generate path points such as:
• x = a + h, y = b
• x = a, y = b + h
• y = b + m(x − a)
• y = b + k(x − a)²
• Custom parametric forms x = x(t), y = y(t)
3. Evaluate f(x, y) for smaller and smaller distances.
4. Compare the last sampled values. If the spread stays within tolerance, the limit is likely consistent.
How to Use This Calculator
- Enter the function using variables x and y.
- Set the target point coordinates a and b.
- Choose an initial step size and refinement count.
- Provide line slopes and parabola coefficients for path testing.
- Optionally add a custom parametric path using t.
- Set a tolerance for deciding path agreement.
- Click Calculate Limit to see the result above the form.
- Review the summary table, chart, and example table.
- Use CSV for spreadsheet review and PDF for printable output.
Frequently Asked Questions
1. What does this calculator estimate?
It estimates the limit of a function of two variables near a chosen point. It uses several approach paths and compares their sampled values numerically.
2. Does matching one path prove the limit exists?
No. A multivariable limit must agree across all nearby paths. One matching path is useful, but it cannot prove existence by itself.
3. Why are slopes and parabolas included?
Different paths can reveal different behaviors. Lines test straight approaches, while parabolas test curved approaches that often expose path-dependent limits.
4. What does tolerance mean here?
Tolerance is the maximum allowed spread between final sampled path values. Smaller tolerance means stricter agreement before calling the limit likely consistent.
5. Can the function be undefined at the target point?
Yes. A limit can still exist even when the function is undefined at the exact point. The calculator focuses on nearby behavior.
6. What functions can I enter?
You can enter expressions using x, y, numbers, parentheses, powers, and common functions like sin, cos, tan, sqrt, log, exp, and abs.
7. Is this a symbolic proof tool?
No. It is a numerical estimator. It helps detect agreement or disagreement across paths, but it does not replace a formal proof.
8. When should I use a custom parametric path?
Use it when you suspect a special approach may behave differently. It is especially helpful for testing nonlinear or problem-specific paths.