2022年数列与函数的极限公式概念 .pdf

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1、极限与连续一、数列的极限定义：1、给定数列，如果当 n 无限增大时，其 通项无限趋过于某个常数A，则称数列 以 A为极限，记作：=A或者(n)2、当数列 以实数 A 为极限时，称数列 收敛于 A，否则称数列 发散。二、数列极限的性质：1)极限的惟一性：若数列收敛，则其极限惟一，若=a，则=a2)有界性：收敛数列必有界.（数列有界是数列收敛的必要非充分条件）3）数列的极限：如数列：,12,432,322,212nn则它的极限为 3 即：3121lim2lim)12(limnnnnnnn三、几个需要记忆的常用数列的极限01l i mnn11l i mnnn0l i mnnq)1(q)(l i m为

2、 常 数aaan四、运算法则：如果AanlimBbnl i m则：BAban)(limBAban)(l i m)0(,limBBAban二、函数极限：?函数极限=A的充分必要条件是=A?函数极限=A的充分必要条件是=A?分段函数极限与该点有无定义无关，只与左右极限有关.即存在=?函数极限的性质：1)极限的惟一性：若函数f(x)当（或）时有极限，则其极限惟一.?极限运算法则：设 limf(x)=A,limg(x)=B,则1)limf(x)=AB 2)limf(x)g(x)=AB 3)当 B时，lim=4)limcf(x)=climf(x)(c为常数)5）limf(x)=limf(x)(k 为常数

3、)?小结：当,时，有=当时当时当时?复合函数运算法则：=?数列的夹逼准则：设有 3 个数列,满足条件：1）（n=1,2,）；2）=a，则数列 收敛，且=a?函数夹逼准则：设函数 f(x),g(x),h(x)在点的某去心邻域内有定义，且满足条件：1）g(x)f(x)h(x);2)=A,.则极限存在且等于 A.?单调有界准则：单调有界数列必有极限.即单调增加有上界的数列必有极限；即单调减少有下界的数列必有极限.?两个重要的极限：?重要极限：=1?重要极限：(1+=e,(1+x=e?无穷小的性质：文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z1

4、0B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J

5、1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G

6、7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7

7、L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1

8、B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2

9、N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码

10、:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N41)有限个无穷小的代数和为无穷小.2)有界变量与无穷小的乘积为无穷小.3)常量与无穷小的乘积为无穷小.4)有极限的量无穷小的乘积为无穷小.5)有限个无穷小的积为无穷小.?在某个自变量变化过程中limf(x)=A 的充要条件是f(x)=A+(x).其中(x)是该自变量变化过

11、程中的无穷小量.?无穷小的比较：设=(x)，=都是自变量同一变化过程中的无穷小.1.若 lim=c(c,是常数)，则称与 是同阶无穷小.2.若 lim=1，则称与 是等价无穷小，记作.3.若 lim=0，则称与 是高阶无穷小，记作=o()4.若 lim=c(c,k 是正整数),则称 与 是 k 阶无穷小.5.的充要条件为-是(或)的高阶无穷小,即或6.,都是自变量同一变化过程中的无穷小,且,lim存在，则有 lim=lim?常用等价无穷小：相乘的无穷小因子可用等价无穷小替换，加、减的不能 x时，x sinx tanx arcsinx arctanx ln(1+x);1-cosx；(1+x-1a

12、x(a);-1xlna(a 0,a);-1 常用等价无穷小：当变量0 x时，21sin,tan,arcsin,arctan,1,ln(1),1cos,2xxxxxxxxx exxxxx-1 11,(1)1 xxxxx?无穷大：函数无穷大无界x时，若 f(x)为无穷大，则为无穷小；x时，若 f(x)为无穷小，且在的某去心邻域内 f(x),则为无穷大.文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N

13、4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:

14、CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10

15、B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1

16、 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7

17、U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L

18、6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B

19、3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4注：分母极限为0，不能用商的运算法则?初等函数：连续函数经过四则运算所得到的函数仍是连续函数.一切初等函数在其定义区间内都是连续的.如果 f(x)是初等函数，是其定义区间内的点，则=f().最值定理：若函数 f(x)在闭区间 a,b上连续，则它在 a,b 上必有最值.有界性定理：若函数 f(x)在闭区间 a,b 上连续，则它在 a,b上有界.介值定理：若函数 f(x)在闭区间 a,b上连续，且 f(a)f(b),则

20、对于 f(a)与 f(b)之间的任何数，在开区间(a,b)内至少存在一点，使得 f()=.零点定理(根的存在性定理)：若函数 f(x)在闭区间 a,b上连续，且 f(a)与 f(b)异号(f(a)f(b)，在开区间(a,b)内至少存在一点，使得 f()=0 求极限：洛必达法则：1、0/0 型：方法：将分子分母分解因式（消去公因子）或者将分子有理化（有理化），再求极限。1、方法：将分子分母同时除以自变量的最高次幂。文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1

21、Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X

22、1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT

23、2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10

24、C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 Z

25、E1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5

26、P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4文档编码:CO1Z10B5X1J1 HT2G7U10C7L6 ZE1B3T5P2N4