关于计算极限的几种方法.doc

目录摘要······················································1 引言······················································2 一. 利用导数定义求极限····································2 ···································2 ·································3 ·······································4 ·········································5 ···············································5 ·················································6 参考文献·················································6 内容摘要摘要: 极限是数学分析中最基本、最重要的概念之一, 极限是微积分的重要基础,研究函数性质的重要手段. 极限是描述函数在无限过程中的变化趋势的重要概念, 本文通过典型例题, 举一反三, 给出几种常用的求极限方法。极限的计算方法很多, 并且有一定的规律和技巧性, 对此, 本文将根据实例进行分析、探讨, 并归纳出一些计算方法. 关键词: 极限;计算;方法 Abstract: the limit is one of the most basic, the most important concept in mathematical analysis, the limit is an important foundation for the calculus, an important means to study the function of the nature of the concept description. The limit is an important trend in the infinite process function, through typical examples, infer other things from one fact,monly used methods for the limits. A lot of calculation method of limit, and there are rules and skills, certain of this, this paper will be based on case analysis, discussion, and sums up some calculation method . Key words: limit; Calculation; methods 引言: 极限是分析数学中最基本的概念之一,用以描述变量在一定的变化过程中的终极状态。早在中载。例如,3 世纪中国数学家刘徽的割圆术,就是用圆内接正多边形周长的极限是圆周长这一思想来近似地计算圆周率的。随着微积分学的诞生,极限作为数学中的一个概念也就明确提出。但最初提出的这一概念是含糊不清的,因此在数学界引起不少争论甚至怀疑。直到 19世纪,由 A.-L. 柯西、 K.(.) 外尔斯特拉斯等人的工作,才将其置于严密的理论基础之上,从而得到举世一致的公认。数学分析中的基本概念的表述,都可以用极限来描述。如函数?? xfy?在 0xx?处导数的定义,定积分的定义,偏导数的定义,二重积分,三重积分的定义,无穷级数收敛的定义,都是用极限来定义的。极限是研究数学分析的基本公具。极限是贯穿数学分析的一条主线。一. 利用导数定义求极限据文?? 1 定理 1 导数的定义:函数)(xf 在 0x 附近有定义,对于任意的 x?, 则)()( 00xfxxfy?????如果x xfxxf xx?????????)()( lim lim 0000存在,则此极限值就称函数)(xf 在点 0x 的导数记为)(' 0xf .即x xfxxfxf x???????)()( lim )(' 000 0在这种方法的运用过程中。首先要选好)(xf ,然后把所求极限。表示成)(xf 在定点 0x 的导数。例1:求 ax xaaxxa aa??? lim 解:原式 0)( lim lim 1 lim 0???????????