Multivariable Limit Calculator
Compute limits of multivariable functions at a point with sampling-based checks, equation preview, concise notes, and responsive results display sections.
Equation Preview
Helping Notes
Enter the function and the point it approaches. The estimate checks many paths; strong agreement suggests the limit exists.
For undefined points, the function may simplify. Try algebraic manipulation such as rationalizing or switching to polar by hand if needed.
Results
Inputs Summary
Estimated Limit
Path Samples
What Is a Multivariable Limit Calculator?
A Multivariable Limit Calculator evaluates limits of functions of several variables at a point by analyzing behavior from every direction. For two variables, we study ; for three variables, . The rigorous definition is: for every there exists such that implies . The calculator automates common strategies—path testing, polar (or spherical) substitution, algebraic simplification, and squeeze arguments—while flagging undefined points and continuity shortcuts. It gives symbolic steps when feasible and high‑precision numerics otherwise, suitable for homework checks, teaching demos, and quick research notes.
About the Multivariable Limit Calculator
Many multivariable limits fail because the value depends on the path taken to the point. To detect this, the tool evaluates along lines , parabolas , or more general curves, comparing resulting limits. For candidate existence, it performs polar substitution about : and examines for independence of . In 3D, spherical coordinates use . The squeeze theorem is applied whenever a clean bound exists. The calculator also contrasts true two‑variable limits with iterated limits (taking one variable at a time), warning when those agree but the genuine limit still fails to exist.
Epsilon–delta (2D):
Polar substitution:
Squeeze:
Path nonexistence test:
Iterated limits caveat:
How to Use This Multivariable Limit Calculator
- Enter the function and the target point (e.g., ).
- Choose tests: algebraic simplification, path checks (lines/parabolas), polar or spherical substitution, squeeze bounds, and iterated limits.
- Compute to see step‑by‑step reasoning, symbolic reductions, and numerical confirmations with clear domain notes.
- Copy the final conclusion: limit value, proof of nonexistence, or “does not exist / undefined”.
Examples
- Nonexistent limit (path dependence): at . Along we get (varies with ) ⇒ no limit.
- Exists and equals 0: . Polar: .
- Squeeze argument: . Since , , hence the limit is 0.
- Iterated vs true limit: For , both iterated limits (first , then , or vice versa) equal 0, but the two‑variable limit does not exist.
Formula Snippets Ready for Rendering
FAQs
How do I prove a multivariable limit exists?
Show path independence (e.g., via polar substitution) and, ideally, provide an epsilon–delta or squeeze bound justifying a single value.
When is polar substitution valid?
Use around a point in 2D. If the resulting expression approaches a value independent of , the limit exists.
Why can iterated limits be misleading?
Taking one variable at a time may hide path dependence. Equal iterated limits do not guarantee the true multivariable limit exists.
What if the function is continuous at the point?
Then the limit equals the function value. Discontinuities (e.g., division by zero) require separate analysis.
How do I show nonexistence quickly?
Find two paths to the point that yield different limit values, such as lines for symmetric rational forms.
Does approaching along curves (not just lines) matter?
Yes. Some functions agree along all lines but differ along parabolas or spirals; the calculator tests several curve families.
Can this handle three variables?
Yes. Use spherical substitution and analogous path tests in to determine limit existence and value.