1、Limit of FunctionOverview2 In this section,we will extend the concepts and results obtained from sequences to functions.defined on a number set N.The limit of a sequence can only be theinfinity.()f nA sequence is a special function limit while n to(positive),its independent variable is x and and may
2、 have various limits.()f x However,for a function can change continuously(x can take any value in its defined interval),The concept of limit of a function3we always use I to express an interval(may be finite or infinite,open,closed or half open).I:an intervalNotation f(x):a functionx:independent var
3、iable()yf x then its independent Function is defined in an interval I,variable x has many changes in several ways:x x x 0 xx0 xx 0 xx The concept of limit of a function4(1)x tends to positive infinity,denoted by;xx tends to negative infinity,denoted by;xx may take positive and negative values,but|x|
4、tends to infinity,denoted by ;xx tends to a finite value x0 only from the right side,denotedby(2)x tends to a finite value x0 only from the left side,denoted by 0;xx 0;xx x may either greater than x0 or less than x0,denotedby 0;xxThe concept of limit of a function5Limit of a function for x tending t
5、o infinity (Limit of a function as x tends to positive infinity)Suppose that A is a constant and the functionIf 0,()0,X such that|()|f xA for all ,xX lim()xf xA,or x We mainly discuss the case cases are similar.and the other two:,)R,fa R.a()f xA as.xthen we say that f(x)has a limit or the limit of f
6、(x)exists as x approaches infinity,and A is called the limit of f(x),denoted byThe concept of limit of a function6xyAA A X0 ,()0X,s.t.|()|f xA holds for all xX.()f xA as x :The geometric meaning of the definitionThe concept of limit of a function7Similarly,it is easy to give the definition of the fo
7、llowing limits:lim()xf xA or()f xA as x and lim()xf xA or()f xA as x .AA A X XAA A 0 ,()0X|()|f xA holds for all x X.x()f xA as:The concept of limit of a function8Remark It is easy to prove the relationship thatlim()lim()lim()xxxf xAf xf xA.Can you prove the above proposition?0,()0X,s.t.|()|f xA hol
8、ds for all|x|X.x()f xA as:0 ,()0X|()|f xA holds for all x X.x()f xA as:The concept of limit of a function10()21f xx3()f xxO1Limit of a function for x tending to a finite value 0 xObserve the behavior of function1.x asThe concept of limit of a function11Since,function()f xso we make a table asIt seem
9、s that while 1x,()3f x.as 1x.do not have definitionx()f x?3.00033.0033.0303.3103.81311.00011.0011.011.11.25()f x2.3132.7102.9702.9972.99970.750.90.990.9990.9999xObserve the behavior of function31()1xf xx ,1x ,at point Limit of a function for x tending to a finite value 0 xThe concept of limit of a f
10、unction12Observe the behavior of functionsin()xf xx We make a table asIt seems that while 0 x,()1f x.as 0.x x()f x?0.999980.998330.958850.8414700.010.10.51.0()f xx0.841470.958850.998330.99998-1-0.5-0.1-0.01Limit of a function for x tending to a finite value 0 xThe concept of limit of a function13Obs
11、erve the behavior of function 1()sinf xx as 0 x.-0.6-0.4-0.20.20.40.6-1-0.50.51Limit of a function for x tending to a finite value 0 xThe concept of limit of a function14x()f x1212Observe the behavior of function,01()1,12xxf xxx as 1.x Limit of a function for x tending to a finite value 0 xThe conce
12、pt of limit of a function15xO()f xObserve the behavior of function21()f xx as 0.x Limit of a function for x tending to a finite value 0 xLimit of a function for x tending to a finite value 0 xDefinition(Neighborhood邻域邻域)The concept of limit of a function16Use 0(,)U x to denote a neighborhood of 0 x,
13、00(,)xx,and 0(,)U x denote the deleted neighborhood去心邻域 of 0 x,0000(,)(,)xxxx,where is any positive real number.0()U x x0 x0 x 0 x The concept of limit of a function17Suppose that the function 0:()Rf U x or 0()R,U x If0,()0,such thatlimit of f(x)as xx0.denoted by 0lim()xxf xA (Limit of f(x)as x tend
14、s to a finite value)RA is a constant.()f xA as 0.xxor Then,we say f(x)has a limit at x=x0,and A is called the0,0 ,such that|()|f xA for all 0().xU x|()|f xA holds for all 0().xU x The concept of limit of a function18xyO)(xfy A A0 x 0 x 0 xAGeometric interpretation of the definition0,0 ,such that|()|
15、f xA for all 0().xU x The concept of limit of a function19Proofwe are looking for a()0,such that 22241xx holds for all x While 1x ,we have 2222(1)1xxx ,so Hence,we need only Then,for all 0 ,()2 ,such that22241xx holds for all Finish.Prove 2122lim4.1xxx For all 0,which satisfies 0|1|.x 22242242|1|.1x
16、xxx|1|.2x 0|1|x.0,0 ,such that|()|f xA for all 0().xU x The concept of limit of a function()0,20Proof Prove 22lim4.xx we are looking for a such that For all 0,24x for all x which satisfies Since,by we can not obtain the required directly,0|2|.x We need to change the left side of this inequality.24|2
17、|2|xxx,holds0,0 ,such that|()|f xA for all 0().xU x The concept of limit of a function21Proof (continued)Prove 22lim4.xx we can restrict to be kept in xa small neighborhood of For example|2|1xIf so,we have|2|5.x ThusSo,2|4|x requires only Therefore,0 ,min 1,5 such that Finish.2,x Because 02x .or 13x
18、.2|4|5|2|xx.5|2|x or|2|5x.20|2|4|5|2|.xxx0,0 ,such that|()|f xA for all 0().xU x 22The concept of limit of a functionOne-Side limits00,xxx()0,such that|()|f xA for all x 0,xx denoted by0lim()xxf xA ,or()f xA as 0 xx.and is A.Similarly,we can define the left limit 左极限左极限,that is 0,If 0,xx In this cas
19、e,we say that when satisfying then A is called the right limit 右极限右极限of f(x)as the right limit of f(x)exists 0lim()xxf xA ,or()f xA as 0 xx.230,0 ,such that|()|f xA for all 00.xxx The concept of limit of a functionOne-Side limits0lim()xxf xA 0lim()xxf xA 0,0 ,such that|()|f xA for all 00.xxx The con
20、cept of limit of a function24x()f x1212Observe the behavior of function,01()1,12xxf xxx as 1.x One-Side limitsThe concept of limit of a function250 xxyO()yf x x0 xyO()yf x This relation shows that,if one of the one-side limits does not exist,or if both of them exists but are not equal,then the limit
21、 of f(x)does not exist.Remark It is easy to prove the relationship that000lim()lim()lim()xxxxxxf xAf xf xA The concept of limit of a function26 Prove the limit of as x tends to 0 does not exist.1arctanx021limarctanxx 021limarctanxx ProofSo,the limit of as x tends to 0 does not exist.1arctanxThe conc
22、ept of limit of a function27Aggregation principle of functional limits lim().nnf xA 0nxxTaking a sequence xn such that ()nf x is also a sequence.,n as0lim(),xxf xA If It isobviously that Conversely,if holds for any sequence xn x0 as n,then we have 0lim()nnxxf xA 0lim().xxf xA The concept of limit of
23、 a function280:()R,f U x 0lim()xxf xA 00()and,nnxU xxx lim().nnf xA Consider a function for any sequence thenAggregation principle of functional limits Remark The corresponding conclusion of the Heines theorem is also suitable for the case x x0+,x x0-,x +,x -and x .The concept of limit of a function
24、29Remark To prove()f x does not have limit at point 0 x is need only find a special sequence nx such that the sequence()nf x is divergent or find two special sequence nx and ny such that two sequence()nf x and()nf y tends to two constant A,B and AB as n.Aggregation principle of functional limits The
25、 concept of limit of a function30Taking the sequence(1)1nxn ,Nn ,(1)0nx as n ,and(1)0nx,we haveAgain,taking(2)122nxn ,Nn ,we haveBy Heines theorem,we have the conclusion follows.Proof(1)lim()limsin0nnnf xn.(2)0nx as n ,and(2)0nx,(2)lim()limsin(2)12nnnf xn .Prove that when 0,x the limit of the functi
26、on does not exist.1()sinf xx The properties of functional limits31(Some properties of functional limits)Suppose that 0lim()xxf xA.(1)(uniqueness)The limit of f(x)must be unique as .0 xx(2)(local boundedness)f(x)is bounded in the deleted neighborhood of x0.()yf x xyOM0 x0 x 0 x That is,0M and 0 such
27、that|()|f xM for all x such that 00|xx.The properties of functional limits32()yf x xyO0 x0 x 0 x (Some properties of functional limits)Suppose that 0lim()xxf xA.(3)(Local preservation of sign),then 0 and a constant 0q such that()0()0)f xqf xq for all x such that 00|xx.0(0)AIf The properties of funct
28、ional limits33()yf x xyO0 x0 x 0 x ()yg x (Some properties of functional limits)Suppose that 0lim()xxf xA.(4)(Local isotone of sign)0lim()xxg xB and 0()()f xg x for allx such that 00|xx,then If such that AB.The properties of functional limits34xyO0 x0 x 0 x ()yg x()yx ()yf x (Some properties of func
29、tional limits)Suppose that 0lim()xxf xA.(5)(Sandwich theorem)0lim()xxg xA and 0 such that()()()f xxg x for all x such that 00|xx,then0lim()xxxA .If The properties of functional limits35()yf x xyO(6)(Monotone bounded criterion)If a function()f x is monotone increasing(decreasing)and bounded above(bel
30、ow)on the interval ,)a,then the limit of the function()f x must exist as x .()yf x axyOAaA(Some properties of functional limits)Suppose that 0lim()xxf xA.Operation rules of functional limits36Theorem(Rational operation rules)Assume that 0lim()xxf xA,0lim()xxg xB,(1)000lim()()lim()lim()xxxxxxf xg xAB
31、f xg x;(2)000lim()()lim()lim()xxxxxxf xg xA Bf xg x;(3)000lim()()lim()lim()xxxxxxf xf xAg xBg x,provided 0B .ThenOperation rules of functional limits37Corollary:If the limit of()f x and()g x both exist as 0 xx,and if a and b are constants,then (1)000lim()()lim()lim()xxxxxxaf xbg xaf xbg x;(2)00lim()
32、lim()nnxxxxf xf x,Nn .Remark The conclusions of the last theorem and its corollary are also correct for the case x x0+,x x0-,x +,x -and x .Operation rules of functional limits38311lim1xxx 21(1)(1)lim1xxxxx 21lim(1)xxx3.Find 311lim.1xxx SolutionAn indeterminate form of type 00Operation rules of funct
33、ional limits39011limxxxx0(11)(11)lim(11)xxxxxxxx 02lim11xxx 02lim(11)xxxxx Find 011lim.xxxxSolution002lim 1lim 1xxxx 1.Operation rules of functional limits400(1)1limnxxx20(1)(1)12!limnxn nnxxxx L L0(1)lim()2!nxn nnxx L L20(1)2!limnxn nnxxxx L L Find 0(1)1lim().nxxnNx Solution.n Operation rules of fu
34、nctional limits41 Find 2112lim().11xxx SolutionAn indeterminate form of type 2112lim()11xxx 211lim1xxx 1.2 11lim1xx Operation rules of functional limitsSuppose that ()()()yfgxf g x is the composition of the functions()yf u and()ug x,and that the composition,fg is defined in a deleted neighborhood of
35、 0 x,0()U x.If 00lim()xxg xu and 0lim()uuf uA.And 00 such that 0()g xu for all 00(,)xU x ,then 00lim()lim()xxxxf g xAf u.420()g xu Question:Why does the theorem need the condition:?(Composition of rational operation rules)Operation rules of functional limits43 Find 2238cos2cos1lim.2coscos1xxxxx An i
36、ndeterminate form of type 00Solutioncos,ux Let1 as.23ux 2238cos2cos1lim2coscos1xxxxx 2212821lim21uuuuu 12(41)(21)lim(21)(1)uuuuu 1241lim1uuu 1212lim(41)lim(1)uuuu 2.By means of the composition ofoperation rule we obtainOperation rules of functional limits1for all 0.xex44Proof(1)We use the definition
37、 of limit.First,we prove 0lim1.xxe Suppose that x 0,0,we require11,xxee and we need onlylog(1).ex Takinglog(1),theneThus0lim1.xxe 0lim1.xxe Second,we proveSuppose that x 0,1.xuueee 0 as0 anduxBy means of the composition of operationrule we obtain00limlimxuxuee 01limuue 1.Prove0000(1)lim1,(2)lim1(0),
38、(3)lim(0).xxxxxxxxeaaaaaOperation rules of functional limits45Proof(continued)(2)Similarly,we can prove 0lim1xxa When a 1.While a=1,000limlim1lim11.xxxxxaWhile 0 a 1.Then,00011limlimlimxxxxxxabb011lim1xxb1.Therefore,we obtain0lim1(0)xxaa Prove0000(1)lim1,(2)lim1(0),(3)lim(0).xxxxxxxxeaaaaaOperation
39、rules of functional limits46Proof(continued)(3)Since00.xx xxaaa Let u=x-x0,then00 as.uxxBy means of the composition of operation rule we obtain 0000limlimxx xxxxxxaaa 000limxx xxxaa 00limxuuaa 0.xa Finish.Prove0000(1)lim1,(2)lim1(0),(3)lim(0).xxxxxxxxeaaaaaOperation rules of functional limitsExample
40、 Suppose that 472lim()0,1xxaxbx determine the constantsSolutionObviously,we know 2lim(),1xxaxbx then 222lim()lim()11xxxxaxaxaxxxexists.So,10,or1,aaand,22lim()lim()11xxxxbaxxxx1.Finish.a and b.Two important limits48OABDCtan xsin x Lets assume that 02x and draw a unit circle.11sintan222xxx.ProofIt is
41、clear thatarea(OAB)area(sector OAB)area(OAC)so that,we havexTwo important limits-10-5510-0.20.20.40.60.8149Proof(continued)we have11sincosxxxBy sandwich theorem,we have Similarly,we can also proof that Then we have the conclusion.sin0 x Since ,while 02x,11sintan222xxx,or sincos1xxx.0sinlim1xxx .0sin
42、lim1xxx .It is easy to prove0limcos1.xx Two important limits50-3-2-1123-20-101020SolutionFinish.Example:Find 0tanlimxxx.Two important limits51 Find 0sinlim(0,0).xmxmnnx0sinlimxmxnxSolution0sinlimxmxmxmxnx g0sinlim()xmmxnmx g.mn Find sinlim(,).sinxmxmNnNnx sinlimsinxmxnx Solution0sin()limsin()tmtmntn
43、 0(1)sin()lim(1)sin()mntmtnt (1).m nmn 0sin()(1)limsin()m ntmtnt Two important limits52-10-55100.10.20.30.40.5SolutionFinish.Find 201coslimxxx.Two important limits53xO1ySolution Let 1arcsintx,so that We can restricting|2t since Hence the variable tThus,Finish.Find 1lim arcsinxxx1sintx.x .can only ap
44、proach 0 as x .Two important limits541lim 1xxexProofso thatSince n as x and nx Suppose that,then 1nxn.11111111nxnnxn ,.Two important limits55xyOeLet tx ,then HenceProof(continued)we have1lim 1xxex.t while x .So1lim 1xxex.Finish.1lim 1xxexTwo important limits56 Find 2lim(1).xxx SolutionLet21,xtthat i
45、s,2,xt and as,so thattx 2lim(1)xxx 21lim(1)ttt 21lim(1)ttt 21lim(1)ttt 2.e 1lim 1xxexReviewThe concepts of limit for a function while independent variable tends to infinityThe concepts of limit for a function while independent variable tends to a valueThe concepts of one-side limitThe properties and operation rules of functional limitsTwo important limits57
