Multivariable Limit Calculator

Calculate limits for functions of two or three variables. Supports different approach paths to determine if a limit exists.

Multivariable Limit Definition: The limit of f(x,y) as (x,y) → (a,b) is L if for every ε > 0, there exists δ > 0 such that if 0 < √((x-a)²+(y-b)²) < δ, then |f(x,y) - L| < ε.

Two Variables (x,y)
Three Variables (x,y,z)
Enter a function of x and y (or x, y, z). Use standard math notation: ^ for exponent, * for multiplication, sin(), cos(), exp(), log(), sqrt(), etc.
(xy)/(x²+y²)
(x²-y²)/(x²+y²)
sin(xy)/(xy)
(x²y)/(x⁴+y²)
(x+y)/(x-y)
√(x²+y²)
(x³+y³)/(x²+y²)
e^(-x²-y²)
Along x-axis
Along y-axis
Along y=x
Along y=mx
Along y=x²
Select different paths to approach the point. If all paths give the same limit, the limit may exist.
Calculating...

Understanding Multivariable Limits

In multivariable calculus, limits become more complex because a point in ℝ² or ℝ³ can be approached along infinitely many paths. For a limit to exist at a point, the function must approach the same value regardless of the path taken.

Mathematical Definition (for two variables):

lim_{(x,y)→(a,b)} f(x,y) = L

if for every ε > 0, there exists δ > 0 such that if 0 < √((x-a)²+(y-b)²) < δ, then |f(x,y) - L| < ε.

Testing for Existence of Limits

1

Path Test: If two different paths to (a,b) yield different limits, then the overall limit does not exist. Common test paths include axes (x=0, y=0), lines (y=mx), parabolas (y=x²), and polar coordinates.

2

Squeeze Theorem: If |f(x,y) - L| ≤ g(x,y) and lim_{(x,y)→(a,b)} g(x,y) = 0, then lim_{(x,y)→(a,b)} f(x,y) = L.

3

Continuity: If f is continuous at (a,b), then the limit exists and equals f(a,b). Polynomials, exponentials, trigonometric functions (where defined), and their sums/products/compositions are continuous on their domains.

Common Limit Problems

Function Limit as (x,y)→(0,0) Exists? Reason
f(x,y) = (x*y)/(x²+y²) 0 along axes, varies along lines No Different limits along y=0 and y=x
f(x,y) = (x²*y)/(x⁴+y²) 0 along lines, 1/2 along y=x² No Different limits along y=mx and y=x²
f(x,y) = sin(x*y)/(x*y) 1 Yes Same limit along all paths (continuous)
f(x,y) = (x³+y³)/(x²+y²) 0 Yes Squeeze theorem with |x³+y³| ≤ (|x|+|y|)(x²+y²)
f(x,y) = (x²-y²)/(x²+y²) 1 along x-axis, -1 along y-axis No Different limits along axes

Polar Coordinates Method

Substituting x = r·cosθ, y = r·sinθ can help analyze limits as (x,y)→(0,0). The limit exists if f(r·cosθ, r·sinθ) → L as r→0, uniformly for all θ.

Example: f(x,y) = (x*y)/(x²+y²)

In polar coordinates: f(r·cosθ, r·sinθ) = (r²·cosθ·sinθ)/(r²) = cosθ·sinθ

As r→0, this depends on θ, so the limit does not exist.

Applications of Multivariable Limits

  • Continuity: Defining continuous functions in higher dimensions
  • Partial Derivatives: Foundation for differentiation of multivariable functions
  • Differentiability: More restrictive than existence of partial derivatives
  • Optimization: Analyzing behavior of functions near critical points
  • Physics & Engineering: Modeling systems with multiple variables

Calculator Features:

  • Supports functions of two or three variables
  • Tests multiple approach paths to determine limit existence
  • Uses symbolic computation with nerdamer.js
  • Visualizes function behavior in 3D with Plotly
  • Provides detailed analysis of each approach path

Frequently Asked Questions

In single-variable calculus, a point can only be approached from two directions (left and right). In multivariable calculus, there are infinitely many paths to approach a point. If different paths yield different limits, the overall limit does not exist. Checking multiple paths helps determine if the limit is path-dependent.

No, finding the same limit along several paths is strong evidence but not proof that the limit exists. There might be some untested path that gives a different limit. To prove a limit exists, you typically need to use the epsilon-delta definition or apply theorems like the Squeeze Theorem.

The calculator uses symbolic algebra to simplify expressions before evaluating limits. For indeterminate forms like 0/0, it attempts algebraic simplification, substitution along specific paths, or conversion to polar coordinates to evaluate the limit along each path.

This calculator can prove a limit does NOT exist by finding two paths with different limits. To prove a limit exists, it provides strong evidence by showing the same limit along multiple paths, but mathematical proof typically requires an epsilon-delta argument or application of a theorem.

"Undefined" typically means the function is not defined at a point (e.g., division by zero). "Limit does not exist" means that as we approach the point, the function values do not approach a single finite number. A function can be undefined at a point but still have a limit as it approaches that point.