Overview
This lecture explains how to determine the limits of functions with several variables, focusing on two main approaches: the two-path test for showing non-existence of limits, and the epsilon-delta definition for proving the existence of limits. The lecture also compares these methods to the one-dimensional case and provides strategies for different types of problems.
One-Variable Limits Review
- For a function of one variable, the limit at a point ( a ) exists if the left-hand and right-hand limits as ( x ) approaches ( a ) are equal and finite.
- In one dimension, a point can only be approached from two directions: the left and the right.
- The process involves checking the limit as ( x ) approaches ( a ) from both sides and confirming they match.
Limits in Several Variables
- For functions of two or more variables (e.g., ( f(x, y) )), a point (such as ( (0, 0) )) can be approached along infinitely many paths, not just two.
- The existence of a limit at a point requires that the function approaches the same value along every possible path to that point.
- Approaches to the point can be along straight lines, curves, or more complex paths, making the analysis more involved than in the one-dimensional case.
- If the limit along any path is different, the overall limit does not exist.
Two-Path Test (Non-Existence)
- The two-path test is used to show that a limit does not exist for a function of several variables.
- To apply the test:
- Choose two different paths approaching the point of interest (e.g., ( (0, 0) )).
- Common choices include straight lines (( y = mx )), parabolas (( y = mx^2 )), or by setting the denominator to zero to find a path.
- Substitute the path equations into the function and compute the limit along each path.
- If the limits along the two paths are different or depend on a parameter (such as ( m )), the limit is not unique and therefore does not exist.
- The two-path test can only prove non-existence of a limit; it cannot confirm that a limit exists, even if the limits along several paths are the same.
- The test is especially useful for "Type 1" problems, where the goal is to check if a limit does not exist.
Epsilon-Delta Definition (Existence)
- To prove that a limit exists for a function of several variables, use the epsilon-delta definition:
- For every ( \epsilon > 0 ), there must exist a ( \delta > 0 ) such that ( |f(x, y) - L| < \epsilon ) whenever ( \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta ).
- The process involves:
- Starting with ( |f(x, y) - L| ) and expressing it in terms of ( x ) and ( y ).
- Using inequalities and substitutions to bound the expression by ( \epsilon ).
- Showing that the bound holds for all possible paths approaching the point.
- If this can be done, the limit exists and equals ( L ).
- This method is required for "Type 2" problems, where the task is to prove the existence of a limit.
Types of Problems
- Type 1: Path-Based Evaluation
- Used to check whether a limit exists by evaluating the function along different paths.
- If the limit depends on the path or is not unique, the limit does not exist.
- Typical approach: set the denominator to zero or use paths like ( y = mx ), ( y = mx^2 ), etc.
- Type 2: Epsilon-Delta Proof
- Used to rigorously prove the existence of a limit.
- Requires starting with the epsilon-delta definition and showing the bound holds for all paths.
- Involves algebraic manipulation and estimation to relate ( |f(x, y) - L| ) to ( \epsilon ).
Key Terms & Definitions
- Limit of Multiple Variables: The value ( L ) that ( f(x, y) ) approaches as ( (x, y) ) approaches ( (x_0, y_0) ).
- Two-Path Test: A method for showing that a limit does not exist by finding two paths with different limit values.
- Epsilon-Delta Definition: The formal definition of a limit, requiring that for every ( \epsilon > 0 ), there is a ( \delta > 0 ) such that ( |f(x, y) - L| < \epsilon ) near the point.
- Unique Limit: The limit value is the same along all possible paths to the point.
- Path: Any way of approaching the point, such as a straight line, curve, or more complex function.
Action Items / Next Steps
- Prepare for the next lecture, which will cover the continuity of functions of two variables.
- Practice distinguishing between Type 1 (path-based) and Type 2 (epsilon-delta) limit problems.
- Review the epsilon-delta definition and work through the provided examples to strengthen understanding.
- Try evaluating limits along different paths and practice constructing epsilon-delta proofs for various functions.