三元函数的三重极限与累次极限

Triple Limits and Iterated Limits of Functions of Three Variables

摘要: 本文梳理三元函数的三重极限、累次极限、混合累次极限的定义，通过实例说明三重极限与累次极限、三重极限与混合累次极限之间不存在互相蕴含关系；给出并证明三重极限与累次极限相等的相关定理及推论，引述已有文献中三重极限与混合累次极限的对应定理与推论。研究结果有助于完善数学分析多元极限知识体系，拓展学生数学思维。

Abstract: This paper sorts out the definitions of triple limits, iterated limits and mixed iterated limits for functions of three variables. Examples are given to illustrate that there is no implication relationship between triple limits and iterated limits, as well as between triple limits and mixed iterated limits. The theorems and corollaries for the equality of triple limits and iterated limits are presented and proved, and the corresponding theorems and corollaries for triple limits and mixed iterated limits from existing literature are cited. The results can enrich the knowledge system of multivariable limits in mathematical analysis and expand students’ mathematical thinking.