Lecture Notes on Limits and Continuity of Multivariable Functions

Introduction

• Focus on Section 13.2: Limits and Continuity of Multivariable Functions.
• Importance of understanding limits before proceeding with multivariable functions.
• Recap on limits from Calc 1: understanding one variable functions as curves on a plane.

Limits of One Variable Functions

• A limit exists if approaching from the left and right leads to the same value.
• In one variable functions, the curve is considered, not just the x-axis.
• Key concept: traveling along the curve determines the limit.

Transition to Multivariable Functions

• Limits in multivariable functions involve more complexity.
• No longer dealing with a single axis but a plane.
• Functions of two variables create surfaces, not curves.
• Challenge: infinite paths to approach a point on a surface.
• Proving a limit exists requires showing all paths lead to the same height.

Proving Limits Do Not Exist

• Easier than proving they do exist.
• Requires showing at least two paths lead to different heights.
• Demonstrated with examples using paths like x=0 or y=0.

Evaluating Limits

• Start by plugging in values.
• If the function is defined and continuous at that point, the limit exists.
• Use techniques like polar coordinates and the squeeze theorem.

Continuity

• A function is continuous at any point inside its domain.
• Domains determine regions of continuity – focus on exclusion of undefined points.
• Composition of functions and continuity: polynomials are continuous everywhere; rational functions everywhere except where the denominator is zero.

Advanced Techniques

• Use of parametric equations to evaluate limits in 3D.
• Squeeze theorem for proving limits exist when standard methods fail.
• Continuous compositions hold continuity properties.

Conclusion

• Key takeaway: understanding and determining limits is crucial for multivariable calculus.
• Techniques taught in the lecture provide tools to approach complex limit problems effectively.