Two Variable Limit Calculator
In two variable limit calculator, enter function type, target point (x,y), and choose an approach path e.g. select “Rational Function” and enter point (0,0). Input function components based on selected type (numerator/denominator for rational functions).
The results will display limit value, approach path analysis, and potential existence conditions.
Function: f(x, y) = x^2 + y^2
Limit as (x, y) → (1, 2): lim (x, y) → (1, 2) x^2 + y^2 = 1^2 + 2^2 = 1 + 4 = 5
Two Variable Limit Calculator
| Function (f(x, y)) | (x, y) → (a, b) | Limit (L) |
|---|---|---|
| f(x, y) = x^2 + y^2 | (x, y) → (0, 0) | 0 |
| f(x, y) = x + y | (x, y) → (1, 2) | 3 |
| f(x, y) = (x * y) / (x^2 + y^2) | (x, y) → (0, 0) | Does not exist |
| f(x, y) = e^(x + y) | (x, y) → (0, 0) | 1 |
| f(x, y) = sin(x * y) | (x, y) → (0, 0) | 0 |
| f(x, y) = (x^2 – y^2) / (x^2 + y^2) | (x, y) → (0, 0) | Does not exist |
| f(x, y) = x^2 + 2 * y^2 | (x, y) → (1, 1) | 3 |
| f(x, y) = ln(1 + x^2 + y^2) | (x, y) → (0, 0) | 0 |
| f(x, y) = (sin(x) * cos(y)) / (x + y) | (x, y) → (0, 0) | Does not exist |
| f(x, y) = sqrt(x^2 + y^2) | (x, y) → (3, 4) | 5 |
Two Variable Limit Formula
lim(x,y)→(a,b) f(x,y)Parameters:
- f(x,y): Two-variable function
- (a,b): Point where limit is evaluated
- Approach Path: Method of reaching point (a,b)
Function Types: Rational: p(x,y)/q(x,y), Polynomial: p(x,y), Exponential: e^(f(x,y))
Approach Methods: Direct Substitution, Along y = mx, Parabolic Path
How to Calculate Two Variable Limit?
To find a two-variable limit, test different approach paths. If all paths yield same value, that’s limit. For example, for f(x,y) = xy/(x² + y²) at (0,0), test paths y = mx and y = x² to verify limit existence.
For function f(x,y) = (x² + y²)/(x + y) approaching (0,0), using direct substitution leads to 0/0 (indeterminate). Along y = x, we get (2x²)/(2x) = x → 0, indicating a potential limit.
Function g(x,y) = xy/(x² + y²) approaching (0,0), different paths give different results. Along y = x we get 1/2, but along y = x² we get 0, proving no limit exists.
With h(x,y) = sin(x²+y²)/(x²+y²) approaching (0,0), all paths yield 1, confirming this as the limit value.
What is Two Variable Limit?
Two Variable Limit examines the behavior of functions f(x,y) as the point (x,y) approaches a specific value (a,b). It’s crucial in multivariable calculus for understanding function continuity and behavior near critical points.