浅析二元函数的重极限与累次极限

Analysing the Double Limit and the Repeated Limit of Binary Function

摘要: 二元函数的重极限是基于一元函数极限的框架搭建而成，而累次极限是基于降维法，将二元函数视为一元函数，依次对相应的变量求极限。这两种极限是同一函数的极限行为，但因其定义方式不同，导致这二者的关系错综复杂。本文首先剖析二元函数的重极限和累次极限的定义及其深层内涵；其次，在典型例题中，根据函数自身的结构特点，详细地讨论了重极限与累次极限的存在性问题，并给出二者的若干种关系。力图使学生深入理解这两种极限并掌握其计算方法，为后续内容的学习打下坚实基础。

Abstract: The double limit of binary function is constructed based on the framework of the limit of unary function. By reducing the dimension, we consider binary function as a unary function and calculate the limit of the corresponding variables in turn. The two kinds of limits are the limiting behavior of the same function, but they are defined in different ways, which makes their relationship complicated. In this paper, firstly the definition and connotation of the double limit and the repeated limit of binary function are analyzed. Secondly, according to the structural characteristics of the function, the existence of the two kinds of limits is discussed in detail, and some relations between them are given. It helps students understand the two kinds of limits in depth and master their calculation, which lays a solid foundation for the follow-up study.