Multivariable Function Limits and Continuity

Multivariable Limits and Neighborhoods

A multivariable function limit evaluates the behavior of a function f(x,y)f(x, y) as the point (x,y)(x, y) approaches a fixed point (x0,y0)(x_0, y_0).

The open disk neighborhood centered at (x0,y0)(x_0, y_0) with radius delta\text{delta} (δ\delta) is defined by the inequality:

(x−x0)2+(y−y0)2<δ\sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta

A limit exists and equals LL, written as:

lim⁡(x,y)→(x0,y0)f(x,y)=L\lim_{(x, y) \to (x_0, y_0)} f(x, y) = L

if for every ϵ>0\epsilon > 0, there exists a corresponding δ>0\delta > 0 such that ∣f(x,y)−L∣<ϵ|f(x, y) - L| < \epsilon whenever 0<(x−x0)2+(y−y0)2<δ0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta

Polar Coordinate Transformations for Limit Evaluation

When evaluating limits at the origin (0,0)(0, 0), converting to polar coordinates simplifies expressions involving quadratic sums:

x−x0=rcos⁡(θ)x - x_0 = r \cos(\theta)

y−y0=rsin⁡(θ)y - y_0 = r \sin(\theta)

Applying the trigonometric identity simplifies the denominator:

r2cos⁡2(θ)+r2sin⁡2(θ)=r2(cos⁡2(θ)+sin⁡2(θ))=r2r^2 \cos^2(\theta) + r^2 \sin^2(\theta) = r^2 (\cos^2(\theta) + \sin^2(\theta)) = r^2

For functions like x3x2+y2\frac{x^3}{x^2 + y^2}, substituting x=rcos⁡(θ)x = r \cos(\theta) and y=rsin⁡(θ)y = r \sin(\theta) gives:

r3cos⁡3(θ)r2=rcos⁡3(θ)\frac{r^3 \cos^3(\theta)}{r^2} = r \cos^3(\theta)

Taking the limit as r→0r \to 0 yields 00 regardless of θ\theta, demonstrating that the limit exists.

Path Dependence and Limit Non-Existence

To test whether a multivariable limit exists, approach the point (0,0)(0, 0) along parameterized linear paths such as y=mxy = m x or general trajectories x(t)x(t) and y(t)y(t).

For the expression x2−y2x2+y2\frac{x^2 - y^2}{x^2 + y^2}, substituting y=mxy = m x yields:

x2−m2x2x2+m2x2=1−m21+m2\frac{x^2 - m^2 x^2}{x^2 + m^2 x^2} = \frac{1 - m^2}{1 + m^2}

Since the result depends on the slope mm, the limit does not exist.

For higher-order rational functions such as x2y2x2y2+(x−y)2\frac{x^2 y^2}{x^2 y^2 + (x - y)^2}, evaluating limits along different approaching paths determines whether the limit is path-dependent or uniquely defined.

Continuity of Multivariable Functions

A multivariable function f(x,y)f(x, y) is continuous at a point (x0,y0)(x_0, y_0) in its domain if the following three conditions are met:

  1. f(x0,y0)f(x_0, y_0) is defined.
  2. lim⁡(x,y)→(x0,y0)f(x,y)\lim_{(x, y) \to (x_0, y_0)} f(x, y) exists.
  3. lim⁡(x,y)→(x0,y0)f(x,y)=f(x0,y0)\lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0).

If a limit fails to exist along any path or does not match the function value at (x0,y0)(x_0, y_0), the function is discontinuous at that point.