뒤로Calc 10: Limits and Continuity of Multivariate Functions
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Multivariate Functions and Level Surfaces
Definition and Examples
Multivariate functions are functions that depend on two or more variables. Their graphs and level surfaces provide geometric insight into their behavior.
Level Surface: For a function , a level surface is the set of points where for some constant .
Example 1: - For , the level surface is the origin. - For , the level surface is a sphere of radius .
Example 2: - For , the surface is a hyperboloid of one sheet. - For , the surface is a cone. - For , the surface is a hyperboloid of two sheets.
Level Curves in Two Variables
Level curves are the set of points where for a fixed . Their shapes depend on the function's form.
Example: Which functions have circles centered at the origin as level curves?
Function | Level Curve Shape |
|---|---|
Hyperbolas (or lines for ) | |
Circles () | |
Not circles | |
Not circles |
Conclusion: Only yields circles as level curves.
Limits and Continuity of Functions of Two Variables
Definition of Limit
The limit of a function as approaches is if, for every , there exists such that:
whenever is in the domain of and .
Limit Laws for Functions of Two Variables
Standard limit laws extend to multivariate functions:
Sum:
Difference:
Constant Multiple:
Product:
Quotient: , provided
Power:
Root: , assuming if is even
Note: Polynomial functions are continuous everywhere.
Examples of Limits
Example 1:
Example 2:
Example 3:
Example 4: - This uses the single-variable result .
Exercise Example
Compute:
Solution: Factor numerator and denominator:
Nonexistence of Limits: The Two-Path Test
Procedure
If a function approaches different values along two distinct paths as , then the limit does not exist.
Reason: If the limit existed, all paths would yield the same value.
Example
Compute:
Path 1 (y-axis): as
Path 2 (x-axis): as
Conclusion: Since the limits along these paths differ, the limit does not exist.
Exercise
Compute:
Answer: The limit does not exist. - Even if horizontal and vertical paths yield the same limit, other paths (e.g., ) may yield different values.
Summary Table: Limit Laws for Multivariate Functions
Operation | Limit Law | Conditions |
|---|---|---|
Sum | None | |
Difference | None | |
Constant Multiple | None | |
Product | None | |
Quotient | ||
Power | None | |
Root | if even |
Additional info: These notes cover foundational concepts in multivariate calculus, including level surfaces, limit definitions, and techniques for determining the existence of limits. The two-path test is a critical tool for analyzing limits in higher dimensions.
