1、Infinitesimal and Infinite QuantitiesInfinitesimal Quantities 无穷小量无穷小量2-1-0.50.51-0.75-0.5-0.250.250.50.7512xsin x1cos x 1020304050-0.2-0.10.11xsin xxan infinitesimal quantity 无穷小量无穷小量 with respect to 0 xx(or x),(or()0 x as 0 xxx)then If a function()x is calledor is called simply infinitesimal 无穷小无穷
2、小.Infinitesimal Quantities3 RemarkDo not confuse an infinitesimal quantity with any small non-zero value.Whether a function is an infinitesimal quantity depends also on how the independent variable changes.0lim()xxf xA Infinitesimal Quantities4isan infinitesimal and()x f(x)A x The necessary and suff
3、icient condition for lim()f xA is()()f xAx,where()x is infinitesimal.Relation of Infinitesimals with limits 0lim()0 xxf xA()()xf xA 0()is an infinitesimal as xxx()()f xAx 0lim()xxf xA Infinitesimal Quantities5Suppose that the independent variables change in the same way.Then (1)The algebraic sum of
4、a finite number of infinitesimals is still an infinitesimal;(2)The product of a finite number of infinitesimals is also an infinitesimal.If a(x)and b(x)are two infinitesimals as x tends to x0,then c(x)=a(x)+b(x)is infinitesimal as x tends to x0;d(x)=a(x)b(x)is infinitesimal as x tends to x0.For exam
5、pleInfinitesimal Quantities6 Suppose that()x is an infinitesimal as 0 xxThen()()x f x is also an infinitesimal as 0.xx and f(x)is a locally bounded function in the deleted neighborhood of0.xf(x)is locally bounded function00such that(),().oMf xMxU x 0()()()as()ox f xMxxU x0()()()()as()oMxx f xMxxU x0
6、lim()()0 xxx f xis an infinitesimal()xsinlim.xxxExample Find Infinitesimal Quantities7The sum or product of two infinitesimal is still an infinitesimal,then how about the quotient of two infinitesimals?Is it also an infinitesimal?For example,as 0,xx,sin x and x2 are both infinitesimals.22sin0,1.xxxx
7、xx We haveThe order of Infinitesimals 8()x()x()0.x (The Order of the Infinitesimals)Assume that and are two infinitesimals as x is varying in a certain way,and()lim0,()xx ()x()x()().xox (1)If then is said to be an infinitesimal of(or is infinitesimal of lower order),denoted byhigher order than()x th
8、an()x()(1)xo ()x In particular,the notation means is an infinitesimal.The order of Infinitesimals 9()x()x()0.x (The Order of the Infinitesimals)Assume that and are two infinitesimals as x is varying in a certain way,and()lim0()xCx ()x()x()().xOx (2)If,then and be two infinitesimals with the same ord
9、er 同阶无穷小同阶无穷小,are said to denoted by The order of Infinitesimals 10()x()x()0.x (The Order of the Infinitesimals)Assume that and are two infinitesimals as x is varying in a certain way,and()lim1()xx ()x()x()().xx (3)If,then and be two equivalent infinitesimals 等价无穷小等价无穷小,are said to denoted by The or
10、der of Infinitesimals 11()x()x()0.x (The Order of the Infinitesimals)Assume that and are two infinitesimals as x is varying in a certain way,and()lim0,()kxCx ()x().x(4)If then is said to be a k-order infinitesimal with respect to In particular,taking 0()xxx,if 00()lim0()kxxxCxx ,then()x is called a
11、k-order infinitesimal as 0 xx.The order of Infinitesimals 120.x Compare the orders for the following infinitesimals as SolutionSince Similarly,it is easy to check that when 0 x,we have2tan,arcsin,arctan,1 cos2xxxxxxxx.43,()2xxx 2()2.xx (1)Since we have4322(2).xxox (2)()sin,xx ().xx Solution0sinlim1,
12、xxx we havesin.xx43200()2limlim()2xxxxxxx 200,2lim2xxx The order of Infinitesimals 13 Prove that when 10,11,.nxxx nNn:Proof:By the formula 10,11nxxxn:0111limnxxxn 11222()(),nnnnnnnabab aabababbL LRationalizing the numerator we obtain 011limnxxx120lim(1)(1)1nnxnnxxxx L L1.n Hence 111.nxxn:Finish.Equi
13、valence transformations of infinitesimals14()()()(),()()()()xxxxxxxx,)()(lim)()(lim)()(lim)()(limthen xxxxxxxxIf()(),()()xxxx and the limit of()()xx exists.()()limlim()()xxxx Theorem Suppose that()x,()x,()x,()x are all infinitesimalsfor a given behaviors of x.()()xx also exists and Then the limit of
14、 ProofSince the conclusion follows from our assumptions.Equivalence transformations of infinitesimals15Solution Find 0.tan3limsin5xxxFinish.tan33xx 00.tan33limlimsin55xxxxxx 3.5 Since we have known that andsin55,xx we haveEquivalence transformations of infinitesimals16SolutionWe have known that By t
15、he last theorem,we haveFinish.Find 32011lim.arctan2xxxx0 x Both the numerator and denominator tend to 0 as.3221arctan2 2,113xxxx.32011limarctan2xxxx2013lim2xxxx 1.6 Equivalence transformations of infinitesimals173300tansinlimlim0 xxxxxxxx.Solution Note:Misuse of the theorem:Find 30.tansinlimxxxx Fin
16、ish.30tansinlimxxxx 2302limxxxx 30tan(1 cos)limxxxx 1.2 Infinite quantities18Definition(Infinite Quantities 无 穷 大 量无 穷 大 量)Let 0:()Rf U x.If 0M,()0M,such that|()|f xM for all x satisfying 00|xx,then()f x is called an infinite quantity as 0 xx,or is called an infinity as 0 xx,denoted by 0lim()xxf x o
17、r()f x as 0 xx.If instead of|()|f xM,we use()f xM (or()f xM )in the above definition,then()f x is called a positive(negative)infinity as 0 xx.Infinite quantities19O()yf x 0 xxyGeometric meaning of an infinite quantityInfinite quantities2031lim,3xx lim(1),xxaa1lim ln(1)xx()sin,f xxx x()f x.x is not a
18、n infinity as oxyunboundedInfinite quantities21(1)If()f x is an infinitesimal but not zero in the required interval,then 1()f x is an infinity;(2)If()f x is an infinity,then 1()f x is an infinitesimal;Theorem Suppose that the independent variables in the following functions vary in the same way.We h
19、ave Note The algebraic sum of two infinities is not necessarily an infinity and the product of an infinity and a bounded function is also not necessarily an infinity.(4)The sum of an infinity and bounded function is also an infinity.(3)The product of a finite number of infinities is also an infinity;Review22The definition of infinitesimal and their orderEquivalence transformations of infinitesimalsInfinite quantities
