Section 4.2 - Limits and Continuity of Two-Variable Functions

Section 4.2 - Limits and Continuity

Limits of Functions of Two Variables

In multivariable calculus, the concept of a limit extends to functions of two variables. Understanding limits is essential for analyzing the behavior of functions near specific points in their domain.

• Definition: Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). The limit of f(x, y) as (x, y) approaches (a, b) is L if: $\lim_{(x, y) \to (a, b)} f(x, y) = L$

• Formal Definition: For every number $\epsilon > 0$, there exists a number $\delta > 0$ such that if $(x, y) \in D$ and $0 < \sqrt{(x - a)^2 + (y - b)^2} < \delta$, then $|f(x, y) - L| < \epsilon$.

• Non-Existence of Limits: If $f(x, y)$ approaches different values along different paths to $(a, b)$, then the limit does not exist.

Example 1

Find $\lim_{(x, y) \to (0, 0)} \frac{3x^2y}{x^2 + y^2}$ along paths $C_1: x = 0$ and $C_2: y = 0$.

• Solution: Along $x = 0$, $f(0, y) = 0$. Along $y = 0$, $f(x, 0) = 0$. Thus, the limit is 0 along both paths.

Example 2

Show that $\lim_{(x, y) \to (0, 0)} \frac{x^2 - y^2}{x^2 + y^2}$ does not exist. (Hint: check limit on paths $C_1: x = 0$ and $C_2: y = 0$.)

• Solution: Along $x = 0$, $f(0, y) = \frac{-y^2}{y^2} = -1$. Along $y = 0$, $f(x, 0) = \frac{x^2}{x^2} = 1$. Since the limits along different paths are not equal, the limit does not exist.

Continuity of Functions of Two Variables

Continuity generalizes to functions of two variables, describing when a function does not have any abrupt changes or jumps at a point.

• Definition: A function f of two variables is continuous at (a, b) if: $\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$

• Continuity on a Domain: f is continuous on D if it is continuous at every point (a, b) in D.

Types of Functions

• Polynomial Function: A sum of terms of the form $c x^m y^n$, where $c$ is a constant and $m, n$ are nonnegative integers. Example: $f(x, y) = 4x^5 + 3x^2y^2 + 6xy^4 - 7y + 6$

• Rational Function: A ratio of polynomials. Example: $g(x, y) = \frac{2xy + 1}{x^2 + y^2}$

• Useful Facts:

  • Polynomial functions are always continuous on $\mathbb{R}^2$.

  • Rational functions are continuous on their domain (where the denominator is not zero).

Example 4

Evaluate $\lim_{(x, y) \to (1, 2)} (x^2y^2 - x^2y + 3x + 2y)$.

• Solution: Substitute $x = 1$, $y = 2$ into the polynomial: $1^2 \cdot 2^2 - 1^2 \cdot 2 + 3 \cdot 1 + 2 \cdot 2 = 4 - 2 + 3 + 4 = 9$.

Continuity of Functions of Three Variables

The concept of continuity extends to functions of three variables.

• Definition: A function f of three variables is continuous at (a, b, c) if: $\lim_{(x, y, z) \to (a, b, c)} f(x, y, z) = f(a, b, c)$

• Polynomial and Rational Functions: The definitions for polynomials and rational functions extend similarly to three variables.

• Useful Facts:

  • Polynomial functions are always continuous on $\mathbb{R}^3$.

  • Rational functions are continuous on their domain.

Example 7

Show that $f(x, y, z) = \frac{1}{x^2 + y^2 + z^2 - 1}$ is continuous at (0, 0, 0).

• Solution: The denominator at (0, 0, 0) is $0^2 + 0^2 + 0^2 - 1 = -1 \neq 0$, so the function is defined and continuous at this point.

Summary Table: Properties of Polynomial and Rational Functions

Function Type	Form	Continuity
Polynomial	$\sum c x^m y^n$ or $\sum c x^m y^n z^p$	Always continuous on $\mathbb{R}^2$ or $\mathbb{R}^3$
Rational	Ratio of polynomials	Continuous on its domain (where denominator $\neq 0$)