Calculator Inputs
Use explicit multiplication such as 2*x*y. Supported functions include sin, cos, tan, abs, sqrt, exp, log, ln, min, max, floor, and ceil.
Example Data Table
This sample shows why path testing matters. For the function below, different paths produce different nearby values at the same target point.
| Example Function | Point | Chosen Path | Observed Expression | Path Value |
|---|---|---|---|---|
| (x² − y²) / (x² + y²) | (0, 0) | y = 0 | (x² − 0) / (x² + 0) | 1 |
| (x² − y²) / (x² + y²) | (0, 0) | x = 0 | (0 − y²) / (0 + y²) | -1 |
| (x² − y²) / (x² + y²) | (0, 0) | y = x | (x² − x²) / (x² + x²) | 0 |
| Conclusion: The limit does not exist because the path values disagree. | ||||
Formula Used
For a two-variable function f(x, y), the calculator examines values of f near the target point (a, b) along several approach paths.
Path rule: choose a path p(h) = (x(h), y(h)) with h → 0.
Directional estimate: Lpath ≈ average of the smallest computed path values.
Tail spread: max(last values) − min(last values).
Agreement spread: max |Lpath − mean of all path estimates|.
If each path stabilizes and the agreement spread stays within tolerance, the calculator reports a likely common limit. This is numerical evidence, not a symbolic proof.
How to Use This Calculator
- Enter the function using x and y with explicit multiplication.
- Set the target point (a, b) where the limit is tested.
- Choose an initial step, refinement levels, shrink factor, and tolerance.
- Optional coefficients let you test custom line, parabola, and absolute-value paths.
- Press Calculate Limit to view the result above the form.
- Read the path table, verdict, and graphs together before concluding.
- Export your findings with the CSV or PDF buttons.
FAQs
1) What does this calculator actually test?
It numerically checks whether several nearby approach paths move toward the same value. Matching paths suggest a likely limit. Disagreeing paths suggest the limit may not exist.
2) Does a matching numerical result prove the limit exists?
No. It provides strong evidence, not a formal proof. A rigorous proof usually needs algebraic simplification, squeeze arguments, polar substitution, or epsilon-delta reasoning.
3) Why are several paths used instead of only one?
A two-variable limit must agree along every possible path. One path can look stable while another shows a completely different behavior. Multiple paths reduce false conclusions.
4) What does tail spread mean?
Tail spread measures how much the last few values change along one path. A small spread suggests that path is settling near a single number.
5) What if the function is undefined at the point itself?
That is common in limit problems. The limit depends on nearby behavior, not necessarily the exact value at the point. The calculator still studies nearby samples.
6) Which expressions are supported?
You can use x, y, numbers, parentheses, powers, and functions such as sin, cos, tan, abs, sqrt, exp, log, ln, min, max, floor, and ceil.
7) When should I reduce the initial step size?
Use a smaller initial step when the surface changes rapidly, the graph looks jagged, or sampled values jump too much. Finer steps often improve numerical stability.
8) Why can the graph look strange near singular points?
Near singularities, values may grow very large or be undefined. Surface plotting hides extreme values to keep the chart readable, but the path table still reveals convergence issues.