Limits of Multivariable Functions: Techniques and Theorems

This lecture covers the limits of functions of several variables, focusing on definitions, examples, and methods for calculating limits. The instructor begins with a recap of previous concepts, emphasizing the importance of understanding limits in multivariable calculus. Various examples are presented, illustrating how to determine the existence of limits using sequences and the epsilon-delta definition. The lecture also introduces polar coordinates as a technique for evaluating limits, demonstrating its effectiveness in simplifying complex expressions. The instructor discusses theorems related to the maximum and minimum of continuous functions, particularly in compact sets, and explains the Bolzano-Weierstrass theorem, which guarantees the existence of convergent subsequences in bounded sequences. The lecture concludes with practical exercises to reinforce the concepts learned, ensuring students grasp the critical aspects of limits in multivariable contexts.