二元函数的二重极限和累次极限关系的探讨

Study on the Relationship between the Double Limit and the Repeated Limit of a Binary Function

摘要: 二元函数的累次极限与二重极限的关系是多元函数微积分学中的一个重点和难点。本文通过五个具体实例，系统归纳出二者之间存在的七种相互关系，进而分析其对后续相关概念的引申影响，为深入理解二元函数的全微分与偏导数、二重积分与累次积分之间的逻辑关系奠定基础。

Abstract: The relationship between the repeated limit and the double limit of a binary function is a key and challenging topic in the calculus of multivariable functions. In this paper, through five concrete examples, we systematically summarize seven possible interrelations between the two types of limits, and further analyze their extended implications for subsequent concepts, thereby laying a foundation for a deeper understanding of the logical relationship between the total differential and partial derivatives of a binary function, as well as the interrelationship between double integrals and iterated integrals.