高等数学教材.docx

目 录 一、函数与极限?································································································?2 1、集合的概念····························································································?2 2、常量与变量····························································································?3 2、函数?·····································································································?3 3、函数的简单性态······················································································?4 4、反函数··································································································?5 5、复合函数·······························································································?5 6、初等函数·······························································································?6 7、双曲函数及反双曲函数·············································································?7 8、数列的极限····························································································?8 9、函数的极限····························································································?9 10、函数极限的运算规则?············································································?11 1 一、函数与极限 1、集合的概念 一般地我们把研究对象统称为元素，把一些元素组成的总体叫集合（简称集）。集合具有确定性（给定集合的元 素必须是确定的）和互异性（给定集合中的元素是互不相同的）。比如“身材较高的人”不能构成集合，因为它的元 素不是确定的。 我们通常用大字拉丁字母?A、B、C、……表示集合，用小写拉丁字母?a、b、c……表示集合中的元素。如果?a?是 集合?A?中的元素，就说?a?属于?A，记作：a∈A，否则就说?a?不属于?A，记作：a A。 ⑴、全体非负整数组成的集合叫做非负整数集（或自然数集）。记作?N ⑵、所有正整数组成的集合叫做正整数集。记作?N+或?N+。 ⑶、全体整数组成的集合叫做整数集。记作?Z。 ⑷、全体有理数组成的集合叫做有理数集。记作?Q。 ⑸、全体实数组成的集合叫做实数集。记作?R。 集合的表示方法 ⑴、列举法：把集合的元素一一列举出来，并用“｛｝”括起来表示集合 ⑵、描述法：用集合所有元素的共同特征来表示集合。 集合间的基本关系 ⑴、子集：一般地，对于两个集合?A、B，如果集合?A?中的任意一个元素都是集合?B?的元素，我们就说?A、B?有 包含关系，称集合?A?为集合?B?的子集，记作?A B（或?B???A）。。 ⑵相等：如何集合?A?是集合?B?的子集，且集合?B?是集合?A?的子集，此时集合?A?中的元素与集合?B?中的元素完全 一样，因此集合?A?与集合?B?相等，记作?A＝B。 ⑶、真子集：如何集合?A?是集合?B?的子集，但存在一个元素属于?B?但不属于?A，我们称集合?A?是集合?B?的真子 集。 ⑷、空集：我们把不含任何元素的集合叫做空集。记作 ，并规定，空集是任何集合的子集。 ⑸、由上述集合之间的基本关系，可以得到下面的结论： ①、任何一个集合是它本身的子集。