Calculator Inputs
Example Data Table
This example shows why path testing matters near a singular point.
| Function | Target | Path | Simplified value | Path limit |
|---|---|---|---|---|
| x^2/(x^2+y^2) | (0, 0) | y = 0 | x^2/x^2 | 1 |
| x^2/(x^2+y^2) | (0, 0) | x = 0 | 0/y^2 | 0 |
| x^2/(x^2+y^2) | (0, 0) | y = x | x^2/(2x^2) | 1/2 |
Formula Used
For two variables, the limit is written as:
lim(x,y)→(a,b) f(x,y) = L
This means f(x,y) becomes close to L whenever (x,y) becomes close to (a,b), from every possible path.
The calculator tests paths by replacing x and y with offsets:
x = a + p(t), y = b + q(t), and t → 0
For three variables, it also uses z = c + r(t). The numeric estimate is the average of the closest finite sample values. The spread is the difference between the largest and smallest tested path estimates. A small spread gives useful numerical evidence. It does not replace a formal proof.
How to Use This Calculator
- Enter the multivariable function using x, y, and optionally z.
- Choose whether the function has two or three variables.
- Enter the target point values a, b, and c.
- Set the largest and smallest t distances.
- Adjust the tolerance when you need stricter path agreement.
- Add custom offsets like t, t^2, sin(t), or 2*t.
- Press the calculate button and read the result above the form.
- Download the summary as CSV or PDF for later review.
Understanding Multivariable Limits
Why multivariable limits are harder
A single variable limit follows one line. A multivariable limit has many paths. The point can be approached from axes, curves, diagonals, and custom directions. This makes the test more delicate. Two paths may give the same value. A third path may show a different value.
What this tool checks
The calculator samples the function near the target point. It uses standard paths and user-defined paths. It then compares values from the smallest distances. When the values become close, the page reports a likely finite limit. When tested paths disagree, the page warns that the limit may not exist.
Why paths matter
Path disagreement is powerful evidence. If one path gives 1 and another gives 0, the limit cannot exist. This is common in rational functions. It often appears when both numerator and denominator become zero at the target. The example table above shows this behavior clearly.
How to read the spread
The spread measures disagreement between near-point estimates. A smaller spread means better agreement. You can lower the tolerance for stricter testing. You can also reduce the smallest t value. Very tiny values may cause floating point noise, so compare results carefully.
Common interpretation
A stable table shows values settling as t moves toward zero. That pattern supports one candidate limit. A table with jumps, blanks, or wide spreads needs caution. It may point to a removable issue. It may also show a genuine failure. Compare the final rows first. Then inspect each path separately.
When to adjust settings
Increase the sample count when the change is slow. Decrease the smallest distance when values still drift. Increase tolerance for rough exploration. Decrease tolerance for stricter comparison. Try negative and positive directions when signs matter. Use custom paths when the denominator suggests a special curve.
Best practice
Use this calculator as a diagnostic helper. Try lines, parabolas, and direction rays. Add custom paths that match the structure of the function. If the tool suggests agreement, follow with algebra, squeeze theorem, polar coordinates, or continuity rules. Numeric evidence is useful, but proof is stronger.
FAQs
1. Can this calculator prove a multivariable limit exists?
No. It gives numerical evidence from several tested paths. A formal proof still needs algebra, squeeze theorem, continuity, or another rigorous method.
2. What does path disagreement mean?
If two valid paths approach the same point but give different limiting values, the multivariable limit does not exist.
3. Which functions are supported?
You can use arithmetic, powers, parentheses, and common functions like sin, cos, tan, sqrt, log, ln, exp, abs, min, max, and pow.
4. How should I enter multiplication?
Use the star symbol. Write 2*x instead of 2x. Write x*y instead of xy. This keeps expression parsing clear and safe.
5. What is the custom path field?
It lets you define offsets from the target point. For example, x offset t and y offset t^2 tests the path (a+t, b+t^2).
6. Why do very small t values sometimes look unstable?
Computer decimals have limited precision. Extremely small distances may create roundoff noise, overflow, or division problems near singular points.
7. Can I calculate three variable limits?
Yes. Choose the three variable option. Then enter z target and optional custom z offset. The page will test three dimensional paths.
8. Why is the result called likely?
Only finitely many paths are tested. Agreement among tested paths is helpful evidence, but infinitely many paths exist near a multivariable point.