预控法在二元函数极限证明中的应用

Application of Pre-Control Method in Proving the Limit of Binary Function

摘要: 多元函数极限是数学分析中的一个重要知识模块，它是在一元函数极限的基础上发展起来的。但因自变量个数增多，在判断多元函数极限的存在与否以及它的求解方法时，比一元函数极限复杂得多。尤其是用定义证明多元函数的极限，这困扰了不少学生。本文以二元函数为例，首先深度剖析二元函数极限的定义。其次，在典型例题的解析中，展示预控法在二元函数极限证明中的具体应用。以期为广大师生用定义证明极限提供一种行之有效的方法，同时帮助学生更好地掌握这部分内容，使其在动手操作中深刻领悟二元函数极限的内涵。这种方法有利于学生养成定量分析的思维习惯，树立勇于克服困难的专业精神。同时也有助于培养学生的逻辑思维能力、数学语言表达能力和分析问题能力。

Abstract: The limit of multivariate function is an important knowledge module in mathematical analysis, and it is developed on the basis of the limit of unary function. However, due to the increase in the number of independent variables, when we judge whether the limit of multivariate function exists or not and how to solve it, it is much more complicated than the limit of unary function. In particular, it troubles many students to prove the limit of multivariate functions by definition. Therefore, taking the binary function as an example, this paper deeply analyzes the definition of binary func-tion firstly. Secondly, the pre-control method is proposed in the analysis of typical examples and its application in proving the limit of binary function is demonstrated. It provides an effective method for the teachers and students for proving the limit by definition, at the same time it helps students to better grasp this content so that they can deeply understand the connotation of the limit of binary function in the hands-on operation. This method is conducive to students to form the habit of quantitative analysis and establish a professional spirit of courage to overcome difficulties. Meanwhile, it also helps to cultivate students’ logical thinking ability, mathematical language expression ability and analytical ability.