Computed Limit Summary
This summary appears above the form after calculation.
Estimated Limit
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Path Spread
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Stable Paths
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Confidence Score
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X Y Z Limit Calculator
Enter a function of x, y, and z. The tool tests several paths near a target point and compares late-stage values.
Plotly Graph
The graph plots f(x,y,z) against the shrinking path parameter t for every tested approach path.
Path Summary Table
| Path | Direction Rule | Last Value | Recent Mean | Local Spread | Status |
|---|---|---|---|---|---|
| Run a calculation to populate this table. | |||||
Detailed Sample Table
| Path | t | x | y | z | f(x,y,z) |
|---|---|---|---|---|---|
| Detailed sample values will appear here. | |||||
Example Data Table
This sample shows how values can settle toward a shared number as the point moves closer to the target.
| Expression | Target Point | Path | Small t | Approximate Value |
|---|---|---|---|---|
| (x+y+z)/(1+x^2+y^2+z^2) | (0,0,0) | (t,t,t) | 0.01 | 0.029991 |
| (x+y+z)/(1+x^2+y^2+z^2) | (0,0,0) | (t,0,0) | 0.01 | 0.009999 |
| (x+y+z)/(1+x^2+y^2+z^2) | (0,0,0) | (0,t,0) | 0.01 | 0.009999 |
Formula Used
This calculator studies a numerical approximation of the multivariable limit lim (x,y,z)→(a,b,c) f(x,y,z).
General path rule:
x = a + αt
y = b + βt
z = c + γt
Then evaluate f(a+αt, b+βt, c+γt) as t → 0.
The calculator tests several paths, including axis paths, diagonal paths, a radial-style path, and one custom path.
If the late-stage values from many paths cluster within the chosen tolerance, the limit is likely consistent near that point. If they disagree strongly, the limit may fail to exist or may need symbolic proof.
How to Use This Calculator
- Enter a valid function using variables x, y, and z.
- Set the target point values a, b, and c.
- Choose the initial step, shrink ratio, and sample count.
- Set a tolerance for comparing path estimates.
- Optionally change α, β, and γ for the custom path.
- Click Calculate Limit to generate results.
- Review the summary, graph, and tables together.
- Export the numerical table as CSV or PDF.
FAQs
1) What does this calculator estimate?
It estimates a three-variable limit numerically. It compares several paths that move toward one target point. Matching path values suggest convergence. Conflicting path values suggest the limit may not exist.
2) Why are several paths tested?
A multivariable limit must approach one value from every direction. Testing only one path can mislead you. Multiple paths give stronger evidence about whether the function settles near one number.
3) Does matching paths prove the limit exists?
No. Matching numerical paths is strong evidence, not a formal proof. A rigorous proof usually needs algebra, inequalities, or a known theorem. Use this tool for exploration and verification support.
4) What if one path returns undefined values?
That often means the function is not defined along that approach, or the expression becomes unstable numerically. The summary will still show which paths stayed valid and which failed.
5) How should I choose the tolerance?
Use a small positive value. Smaller tolerances demand tighter agreement. For many classroom examples, 0.01 or 0.001 works well. Very tiny tolerances can exaggerate rounding noise.
6) What does the shrink ratio control?
It controls how fast t moves toward zero. With a ratio of 0.5, each step is half the previous step. Smaller ratios approach the target faster and can reveal late-stage behavior better.
7) Can I use trigonometric or exponential functions?
Yes. You can use common functions such as sin, cos, tan, sqrt, exp, and log. Write them in standard mathematical form inside the function field.
8) Why should I inspect the graph and tables together?
The graph shows trend behavior quickly. The tables show exact sampled values and path spreads. Together they help you judge stability, detect anomalies, and compare approach directions more carefully.