Lecture on Limits and Continuity of Multivariable Functions

Introduction

• Focus on understanding the concept of limits in multivariable functions.
• Importance of understanding the concept rather than just memorizing rules.

Limits in Single Variable Functions

• In single-variable functions, limits exist if they equal a number.
• Limits do not exist if they equal an asymptote or have different values from the left and right.
• Limits are determined by the height of the function as you approach a value from both sides along the curve.

Multivariable Functions

• Linearity in one variable: 1D curve in 2D space.
• Multivariable functions involve multiple independent variables and surfaces in higher dimensions.
• The limit exists if it equals a certain value from all paths approaching a point.

Evaluating Limits in Multivariable Functions

• For two variables: view the function as a surface in 3D.
• Problems arise due to infinite paths possible to approach a point.
• Limit does not exist if two paths yield different heights (values) at the same point.

Techniques for Evaluating Limits

1. Plug-In Method:
   • Always start by plugging in the numbers.
   • If it results in an indeterminate form, further analysis is needed.
2. Path Methodology:
   • Use simple paths like setting x = 0 or y = 0 to test for limits.
   • If paths yield different limits, the limit doesn’t exist.
3. Polar Coordinates & Algebraic Manipulation:
   • Use polar transformations for easier evaluation when dealing with expressions like x² + y².
4. Squeeze Theorem:
   • Used to prove limits when direct substitution and path testing are inconclusive.

Continuity

• A function is continuous at a point if it is defined in a neighborhood around that point.
• A function is continuous on its domain, where there are no holes, jumps, or undefined points.

Continuity of Multivariable Functions

• Continuous on a domain where the function is defined.
• Check for issues like division by zero, undefined points, and restrictions (such as inside roots).

Composition and Continuity

• Compositions of continuous functions are continuous.
• Domain restrictions of composed functions are determined by both the outer and inner functions.

Continuity and Domains

• Polynomial functions are continuous everywhere.
• Rational functions are continuous where the denominator is not zero.
• Use domain to find continuity.

Advanced Techniques

• For functions with three independent variables, parametric curves help in evaluating limits.
• The transition from 2D to 3D path consideration.

Key Takeaways

• Master the fundamental understanding of limits for further applications in calculus.
• Recognize the distinction between proving limits exist vs. proving they do not.
• Use a systematic approach: plug in, path analysis, polar transformation, and the squeeze theorem as tools for evaluation.

This lecture provides a thorough understanding of evaluating limits and understanding continuity in multivariable calculus, essential for further studies in calculus and mathematical analysis.