Multivariable Limit Evaluator
Analyze limits of functions f(x,y) as they approach a point.
Supported: sin, cos, sqrt, ^, etc.
Path Analysis
Surface Visualization z = f(x,y)
Drag to RotateNote: The graph shows the function behavior around the target point (red dot).
Understanding Multivariable Limits
In single-variable calculus, a limit lim x→c f(x) exists if approaching c from the left and right gives the same value. However, in the world of Multivariable Calculus, things get much more interesting (and tricky!).
For a function f(x, y), approaching a point (a, b) isn’t limited to just two directions. You can approach that point from infinite directions: vertically, horizontally, diagonally, or even spiraling in along a curve.
Common Techniques for Evaluation
- Direct Substitution: Always try plugging the numbers in first. If you get a real number (like 5 or -2), that’s usually your answer because most standard functions are continuous.
- Factoring & Simplifying: If you get 0/0, try factoring the numerator and denominator. Often terms will cancel out, revealing the true limit.
- The Two-Path Test: This is what this tool demonstrates. We check the limit along x = 0 (the y-axis), y = 0 (the x-axis), and lines like y = x. If any two paths give different numbers, the limit doesn’t exist.
Frequently Asked Questions (FAQ)
Q: Why does the calculator show “Limit Likely Exists”?
A: Numerical tools can check many paths, but they cannot check every infinite path. If all checked paths agree, the limit likely exists, but a formal mathematical proof (epsilon-delta) is required to be 100% sure.
Q: What does “Indeterminate Form” mean?
A: It typically refers to a result like 0/0 or ∞/∞. This doesn’t mean the limit is undefined; it means more work is needed (like factoring or using L’Hôpital’s rule) to find the value.
Q: Can I use polar coordinates?
A: Yes! Converting to polar coordinates (x = r·cosθ, y = r·sinθ) is a powerful method. If the limit depends on θ as r→0, the limit does not exist.