Advanced Calculator
Example Data Table
| Function | Point | Path Evidence | Expected Result |
|---|---|---|---|
(x*y)/(x^2+y^2) |
(0, 0) | y=x gives 1/2. y=-x gives -1/2. |
Limit does not exist. |
(x^2*y)/(x^2+y^2) |
(0, 0) | Many line paths approach 0. | Likely 0. |
x^2+y^2 |
(0, 0) | Radial paths approach 0. | Limit is 0. |
sin(x+y)/(x+y) |
(0, 0) | Non-singular nearby paths approach 1. | Likely 1. |
Formula Used
The two variable limit is written as
lim(x,y)→(a,b) f(x,y) = L.
It means that f(x,y) becomes close to L
when (x,y) becomes close to (a,b).
This calculator uses path testing. For a line path,
y-b=m(x-a). For a vertical path,
x=a. For a radial path,
x=a+r cos θ and y=b+r sin θ.
For a parabolic path, y-b=k(x-a)^2.
If many paths approach the same value, the result is marked likely stable. If different paths approach different values, the limit likely does not exist. A formal solution still needs algebraic proof.
How to Use This Calculator
- Enter a function that uses
xandy. - Enter the target point
(a,b). - Set the starting distance and shrink factor.
- Add line slopes and parabolic coefficients if needed.
- Press the calculate button.
- Read the summary above the form.
- Download the result as CSV or PDF.
Supported functions include sin, cos, tan,
sqrt, abs, exp, log,
ln, pow, min, and max.
Understanding Limits of Two Variables
Why Paths Matter
A limit of two variables is harder than a single variable limit. There are many ways to approach one point. A path may move straight, curve, spiral, or follow another rule. The limit exists only when all valid approaches lead to one value. This is why path testing is useful. It can quickly expose disagreement between routes.
Numerical Evidence
This calculator samples several common paths. It moves closer to the target point step by step. Each step uses a smaller distance. The final values are compared across paths. The tail spread shows how much the last values change. A small tail spread suggests local stability. A large spread suggests movement or oscillation.
When a Limit Fails
A limit fails when two paths lead to different values. For example, one line may approach one half. Another line may approach negative one half. Then no single value can describe the function near the point. The calculator reports this by comparing final path values. It also marks paths that remain unstable.
Use With Algebra
Numerical testing is a strong guide. It is not a final proof. Use it before writing a formal solution. After finding a likely answer, confirm it with algebra. Common proof tools include factoring, squeezing, polar substitution, and inequalities. When the calculator shows disagreement, search for a simple path proof. When it shows agreement, try to bound the expression. This workflow saves time and improves accuracy.
FAQs
1. What is a limit of two variables?
A limit of two variables describes the value a function approaches as x and y move toward a chosen point together.
2. Why does this calculator test paths?
Different paths can approach the same point. If those paths produce different values, the limit does not exist.
3. Can numerical testing prove a limit?
No. Numerical testing gives evidence. A complete proof usually needs algebra, inequalities, polar form, or the squeeze theorem.
4. What does tail spread mean?
Tail spread is the difference between the last sampled values on a path. Smaller spread suggests stronger convergence.
5. What should I enter for slopes?
Enter comma separated values like -2, -1, 0, 1, 2. These test straight line approaches.
6. What are parabolic paths?
Parabolic paths follow a curve such as y-b=k(x-a)². They can reveal failures missed by straight lines.
7. Why do some samples show undefined?
The function may divide by zero or leave its domain at sampled points. Try different paths or smaller distances.
8. Which functions are supported?
The calculator supports common functions such as sin, cos, tan, sqrt, abs, exp, log, ln, pow, min, and max.