浅谈数学分析中求极限的常用方法Preliminary analysis on the common method of limit problem in
mathematical analysis
摘要
求极限问题是数学分析学习的基础，也是其极为重要的内容之一。极限问题分为函数极限和数列极限两类，其他很多重要的数学概念的学习都建立在极限基础上，比如导数，积分，级数等等。因此要学好数学分析，就要学好极限。解决极限问题看似简单，但却很抽象，往往很难求出。我们不能仅仅局限于用极限的概念求极限，我们应该掌握多种方法，并且运用各种方法结合，快速而准确的求出极限。因为极限贯穿于数学分析学习的始终，许多数学概念是从极限出发而得出的。所以反过来，我们也可以通过有关于极限的数学概念而求出极限。但是这并不是非常容易的事情，因为极限问题过于抽象，所以我们应该单独的学习各种方法针对性的求极限，最后再进行整合，把多种方法相结合来求极限。由此可以看出求极限问题是十分繁琐的，针对这种情况，本文中介绍了多种基本的求极限方法和注意事项，并且通过例题的运算过程清晰明了的展现了极限问题的解决过程，使极限问题变得相对简单易懂，为数学分析的学习打下基础。
关键词：数列极限；函数极限；方法
Preliminary analysis on the common method of limit problem in
mathematical analysis
Abstract
Limit problem is the base of mathematical analysis. It can be divided into function limit and sequence limit, both of them are very important. Mary other important mathematical ideas are based on limit, such as derivative integral and progression. If one wants to learn mathematical analysis well, he must learn limit well. It is usually very hard to solve limit problem, it seems to be simple, but rather abstract in fact we can not be restricted to solve limit problem by using the concept of limit. We should master multiple methods and use them together to solve the limit problem quickly and accurately. Limit exists in the whole process of mathematical analysis many mathematical concepts start from limit. On the contrary, we can use these concepts to solve limit problem. All these are no easy things. Because of the abstract of limit problem, we should learn multiple of methods in a target way and eventually combine them to solve limit problem. We can see that solving limit problem is very complicated. Aiming at this circumstances, this article introduce multiple basic ways to solve the problem and master needing attention, The calculation of example shows the solving process of limit problem. It make limit problem easier to understand and provide a foothold for the study of mathematical analysis.
目录
摘要 ................................................................................................................................ I Abstract ................................................................................................................................ III 引言 . (1)
1 极限相关的概念 (2)
1.1 数列极限 (2)
1.2 函数极限 (2)
1.3 函数极限和数列极限的关系 (3)
2 求极限的常用方法 (4)
2.1 极限的四则运算法则 (4)
2.2 两个重要极限 (5)
2.3 用函数的连续性求极限 (7)
2.4 等价无穷小代换 (8)
2.5 洛必达法则 (9)
2.6 根据定积分的定义求极限 (11)
2.7 利用泰勒公式求极限 (12)
2.8 利用极限存在准则求极限 (13)
2.9 拉格朗日中值定理求极限 (15)
3 求极限的小技巧 (15)
3.1 有界函数与一个无穷小量的积仍为无穷小量 (16)
3.2 换元法 (16)
3.3 数列极限转化成函数极限 (17)
结论 (18)
参考文献 (19)
