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2021年高等数学电子版.pdf

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1、.-.word.zl-第一章极限与连续第一节数列的极限一、数列极限的概念按照某一法则，对于每一个Nn，对应一个确定的实数nx，将这些实数按下标n从小到大排列，得到一个序列,21nxxx称为数列，简记为数列nx，nx称为数列的一般项。例如：,1,43,32,21nn,2,8,4,2n,21,81,41,21n,)1(,1,1,11n,)1(,56,43,34,21,21nnn一般项分别为1nn，n2，n21，1)1(n，nnn 1)1(数列nx可看成自变量取正整数n的函数，即)(nfxn，Nn设数列nnxnn1)1(，来说明数列nx以 1为极限。为使100111)1(|1|1nnnxnn，只

2、需要100n，即从101 项以后各项都满足1001|1|nx，为使100000111)1(|1|1nnnxnn，只需要100000n，即从 100001项以后各项都满足1000001|1|nx，为使nnnxnn11)1(|1|1（是任意给定的小正数），只需要1n，即当1n以后，各项都满足|1|nx。令1N，当Nn时，1n，因此有|1|nx，即任意给定小正数，总存在正整数1N，当Nn时的一切nx都满足|1|nx，则定义：设nx为一数列，如果存在常数a，对于任意给定的正数（不论它多么小），总存在正整数N，使得当Nn时的一切nx都满足不等式|axn则说常数a是数列nx的极限，或者说

3、数列nx收敛于a，记为axnnlim或axn)(n如果不存在这样的常数a，则说数列nx没有极限，或者说数列nx发散。|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 1 页，共 17 页.-.word.zl-数列nx以a为极限的几何意义：任意给定的正数，总存在正整数N，当Nn时的一切nx，有|axn即axan或),(aaxn也就是当Nn的一切nx都落在a的邻域),(aU，在),(aU的外边至多有N项（图）1xNxa1Nxa2Nxa例 1 证明数列,1,43,32,21nn的极限为1。证明：分析：为使11|nnaxn，只需要11n，或11n，即11n

4、证明：任意给定小正数，取11N，当Nn时的一切nx满足1111|1|nnnxn因此，11limnnn例 2 已知2)1()1(nxnn，证明数列nx的极限是0。分析：为使0)1()1(|2naxnn，只需要2)1(1n，由于11)1(1)1(122nnn，故11n时，即11n，或11n时2)1(1n。证明：任意给定小正数，取 11N，当Nn时的一切nx满足11)1(10)1()1(|0|2nnnxnn因此，0)1()1(lim2nnn例 3 设1|q，证明等比数列,112nqqq的极限是0。证明：任给0（设0），由于11|0|0|nnnqqx为使|0|nx，只需11|0|nnqq，解得ln|l

5、n)1(qn，或|lnln1qn。故取|lnln1qN，当Nn时，有11|0|0|nnnqqx因此，0lim1nnq。二、收敛数列的性质|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 2 页，共 17 页文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:C

6、L4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7

7、W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10

8、 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8

9、N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3

10、 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7

11、Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8文档编码:CL4Q8N7W10M10 HP4H8N5K9L3 ZU1V7Q4S1U8

13、界的；否则，则说数列nx是无界的。定理 2（收敛数列的有界性）如果数列nx有极限，则数列nx一定有界。证明：注意到|aaxaaxxnnn，可证明定理2。定理 3（收敛数列的保号性）如果axnnlim，且0a（或0a），则存在正整数N，当Nn时的一切nx，有0nx（或0nx）。证明：取2a即可证明定理。推论如果数列nx从某项起有0nx（或0nx），且axnnlim，则0a（或0a）。对于数列nx，从中抽取1nx，2nx，knx，称为数列nx的一个子数列。定理 4 如果数列nx收敛于a，则数列nx的任何子数列都收敛，且收敛于a。第二节函数的极限一、函数极限的定义1自变量趋向于无穷大时函数的极限数

14、列是特殊的函数，如1)(nnnfxn，,2,1n，且n时，1nx，考虑函数1)(xxxfy，是否有x时，1)(xf？任意给定小正数，为使|11|1)(|xxxf，只要|11|x，即1|1|x。由于1|1|xx，即11|x即可。任给0，存在正数11X，当Xx|时，对应的函数值)(xf满足|11|1)(|xxxf即当x时，)(xf以 1为极限。定义 1 设函数)(xf当|x大于某一正数时有定义。如果存在常数A，对于任意给定的正数（不论|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 3 页，共 17 页文档编码:CB5S

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16、0T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6

17、R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB

18、5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6

19、S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5

20、W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:

21、CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9.-.word.zl-它多么小），总存在正数X，使得x满足不等式Xx|时，对应函数值)(xf满足|)(|Axf则说常数A为函数)(xf当x时的极限，记为Axfx)(lim或Axf)(（当x）Axfx)(lim：0，0X，当Xx|时，|)(|Axf。例 1 证明03limxx。分析：

22、为使|03|x，只要|3|x，即|3x，或3|x。证明：0，3X，当Xx|时，|3|03|xx，因此03limxx。Axfx)(lim的几何解释：0，0X，当Xx|时，|)(|Axf即Axf)(或AxfA)(如图所示：如果0，0X，当Xx时，|)(|Axf，则说x时，Axf)(，记为Axfx)(lim；如果0，0X，当Xx时，|)(|Axf，则说x时，Axf)(，记为Axfx)(lim显然，Axfx)(limAxfx)(lim，Axfx)(lim例如：xxxf|)(，有1)(limxfx，1)(limxfx。2自变量趋向于有限值时函数的极限例 1，12)(xxf，2x时，5)(xf；例 2：1

23、1)(2xxxf，定义域为1x，但1x时，2)(xf；任意给定小正数，为使|42|512|)(|xxAxf，只要|2|2 x，即2|2|x即可。任意给定小正数，为使|21)1)(1(|211|)(|2xxxxxAxf只要|1|x，即|1|0 x即可。定义 2 设函数)(xf在点0 x的某一去心邻域内有定义。如果存在常数A，对于任意给定的正数（不论它多么小），总存在正数，使得x满足不等式|00 xx时，对应函数值)(xf满足|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 4 页，共 17 页文档编码:CB5S5P2R4G3 HU6S

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28、5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码

29、:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3

30、HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9.-.word.zl-|)(|Axf则说常数A为函数)(xf当0 xx时的极限，记为Axfxx)(lim0或Axf)(（当0 xx）Axfxx)(lim0：0，0，当|00 xx时，|)(|Axf。例 2 证明8)13(lim3xx。分析：为使|93|8)13(|xx，只要|3|3 x，即3|3|x。证明

31、：0，取3，当|3|0 x时，对应函数值满足|3|3|8)13(|8)(|xxxf因此，8)13(lim3xx。Axfxx)(lim0的几何解释：0，0，当|00 xx时，|)(|Axf即Axf)(或AxfA)(即),(00 xUx时，),()(AUxf如图所示：如果0，0，当0 xx时，|)(|Axf，则说x从0 x的右侧趋向于0 x（记为0 xx）时，Axf)(，记为Axfxx)(lim0，或Axf)(0；如果0，0，当xx0时，|)(|Axf，则说x从0 x的左侧趋向于0 x（记为0 xx）时，Axf)(，记为Axfxx)(lim0，或Axf)(0；显然，Axfxx)(lim0Axfxx

32、)(lim0，Axfxx)(lim0例 3 设函数0,10,00,1)(xxxxxxf当0 x时，)(xf的极限不存在。例 4 证明ccxx0lim例 5 证明00limxxxx例 6 证明424lim22xxx例 7 证明0sinlimxxx二、函数极限的性质定理 1（函数极限的唯一性）如果)(lim0 xfxx存在，则极限是唯一的。定理 2（函数极限的局部有界性）如果Axfxx)(lim0，则存在正数M和，使得当|00 xx|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 5 页，共 17 页文档编码:CB5S5P2R4G3 HU6S10T1U

33、2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5

34、J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P

35、2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T

36、1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2

37、D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S

38、5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S1

39、0T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9.-.word.zl-时，有Mxf|)(|。证明：|)(|)(|)(|AAAxfAAxfxf定理 3（函数极限的局部保号性）如果Axfxx)(lim0，且0A（或0A），则存在常数0，使得当|00 xx时，有0)(xf（或0)(xf）。推论如果在0 x的某去心邻域),(00 xU，0)(xf（或0)(xf），且Ax

41、函数)(xf称为当0 xx（或x）时的无穷小。例如：0)1(lim1xn，因此)1(x为1x时的无穷小；01limxn，因此x1为x时的无穷小。)(xf为0 xx时的无穷小0)(lim0 xfxn0，0，当|00 xx时，|)(|xf；)(xf为x时的无穷小0)(limxfn0，0X，当Xx|时，|)(|xf；定理 1 在自变量的同一变化过程0 xx（或x）中，函数)(xf以A为极限的充分必要条件是Axf)(，其中是无穷小。证明：必要性：设Axfxn)(lim0，则0，0，当|00 xx时，|)(|Axf。令Axf)(，则是0 xx时的无穷小，且Axf)(。充分性：设Axf)(，其中A

42、为常数，是0 xx时的无穷小。于是，0，0，当|00 xx时，|，即|)(|Axf，因此，A为)(xf当0 xx时的极限，或Axfxn)(lim0。二、无穷大如果当0 xx（或x）时，对应的函数值的绝对值|)(|xf无限增大，则称函数)(xf为0 xx（或x）时的无穷大。定义 2 设函数)(xf在0 x的某一去心邻域内有定义（或|x大于某一正数时有定义）如果对于任意给定的正数M（不论它多么大），总存在正数（或正数X），当x满足|00 xx（或Xx|）时，|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 6 页，共 17 页文档编码:CB5S

43、5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S1

44、0T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6

45、R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB

46、5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6

47、S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5

48、W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:

49、CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9文档编码:CB5S5P2R4G3 HU6S10T1U2Y4 ZJ5W6R2D5J9.-.word.zl-对应函数值)(xf满足Mxf|)(|则说函数)(xf为0 xx（或x）时的无穷大。如果函数)(xf为0 xx（或x）时的无穷大，也可记为)(lim0 xfxn（或)(limxfn）例如：11x为1x时的无穷大；12x为x时的无穷大。)(lim0 xfxn：0M

50、，0，当|00 xx时，Mxf)(；)(limxfn：0M，0X，当Xx|时，Mxf)(。如果)(lim0 xfxn，则直线0 xx是函数)(xfy的图形的铅直渐近线；如果Axfn)(lim，则直线Ay是函数)(xfy的图形的水平渐近线。定理 2 在自变量的同一变化过程中，如果)(xf为无穷大，则)(1xf为无穷小；反之，如果)(xf为无穷小，且0)(xf，则)(1xf为无穷大。第四节极限运算法则定理 1 有限个无穷小的和也是无穷小。证明：以两个无穷小的和为例：设及是0 xx时的两个无穷小，令。由于是0 xx时无穷小：0，01，当10|0 xx时，2|；又由于是0 xx时无穷小：对于0，02，

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