LIMITS OF SEQUENCES AND FUNCTIONS

Transcription

1 ФЕДЕРАЛЬНОЕ АГЕНТСТВО ПО ОБРАЗОВАНИЮ Государственное образовательное учреждение высшего профессионального образования «ТОМСКИЙ ПОЛИТЕХНИЧЕСКИЙ УНИВЕРСИТЕТ» VV Koev LIMITS OF SEQUENCES AND FUNCTIONS TeBook Рекомендовано в качестве учебного пособия Редакционно-издательским советом Томского политехнического университета Издательство Томского политехнического университета 9

2 UDС 7 VV Koev Limis of Sequeces ad Fucios Tebook The secod ediio Tomsk, TPU Press, 9, pp The ebook cosiss of wo pars devoed o he mahemaical coceps of Limis Differe formulaios of is help o udersad beer he uiy of various approaches o his cocep The basic coceps are eplaied by eamples ad illusraed by figures The ebook is desiged for Eglish speakig sudes Reviewed by: VA Kili, Professor of he Higher Mahemaics Deparme, TPU, DSc -9 Koev VV -9 Tomsk Polyechic Uiversiy

3 Preface Every sude of a echical uiversiy has o be well grouded i mahemaics o sudy egieerig sciece whose mahemaical ools are based o Calculus The cocep of he i is esseial for calculus I is impossible o overesimae he imporace of his cocep for moder sciece I was a very grea advace o all former achievemes of mahemaics Limis epress he coceps of ifiie small ad ifiie large quaiies i mahemaical erms The comprehesio of is creaes he ecessary prerequisies for udersadig oher coceps i Differeial Calculus ad Iegral Calculus such as derivaives, defiie iegrals, series, ad solvig differe problems: calculaio of he area of a figure, he legh of a arc of a curve, ad so o This ebook is ieded for sudes sudyig mehods of higher mahemaics I covers such coe areas as Limis of Sequeces, Basic Elemeary Fucios, ad Limis of Fucios Each par of he ebook coais basic mahemaical cocepios There are preseed differe formulaios of is o demosrae he uiy of various approaches o his cocep Iuiive argumes are combied wih rigorous proofs of proposiios May useful eamples ad eercises are eplaied ad illusraed graphically The book is useful for sudes specialized i differe areas of eperise o broade ad mehodize a kowledge of he basic mahemaical mehods I ca be also used by eachers i he classroom wih a group of sudes I hak Professor Vicor A Kili, who has made useful cosrucive suggesio abou he e His careful work i checkig he e helped me o avoid may iaccuracies The auhor welcomes reader s suggesios for improveme of fuure ediios of his ebook

4 CONTENTS Preface Coes Chaper NUMERICAL SEQUENCES Basic Defiiios 6 Bouded Sequeces 7 Moooe Icreasig Sequeces 8 Illusraios 9 Limis of Numerical Sequeces Ifiiesimal Sequeces Ifiie Large Sequeces 6 Properies of Ifiiesimal Sequeces 8 Properies of Limis of Sequeces Classificaio of Ifiiesimal Sequeces 6 Compariso Bewee Ifiiesimal Sequeces 7 Classificaio of Ifiie Large Sequeces 8 Compariso bewee Ifiie Large Sequeces 9 Theorems of Sequeces 7 Number e 8 Chaper FUNCTIONS Elemeary Fucios: Shor Review Power Fucios y Epoeial Fucios y a Logarihmic Fucios y log Trigoomeric Fucios 7 Iverse Trigoomeric Fucios 6 Hyperbolic Fucios a

5 Limis of Fucios 7 Preiary Discussio 7 Basic Cocepios ad Defiiios 8 Illusraios Oe-Sided Limis Properies of Ifiiesimal Fucios Properies of Limis of Fucios Eamples 6 6 Classificaio of Ifiiesimal Fucios 9 7 Compariso Bewee Ifiiesimal Fucios 6 8 Classificaio of Ifiie Large Fucios 6 9 Compariso Bewee Ifiie Large Fucios 6 The Mos Impora Limis 66 Theorem 66 Calculus of Approimaios 7 Theorem 7 Theorem 76 Theorem 79 Theorem 8 Calculus of Approimaios 8 Summary: Ifiiesimal Aalysis 87 Table of Ofe Used Equivale Ifiiesimal Fucios 88 Review Eercises 89 6 Coiuiy of Fucios 9 6 Basic Defiiios 9 6 Properies of Coiuous Fucios 9 7 Seleced Soluios 96 8 Seleced Aswers 98 Refereces 99

6 Sequeces NUMERICAL SEQUENCES Basic Defiiios The mahemaic cocep of a sequece correspods o our ordiary oio abou a sequece of eves i a sese of a cerai order of eves A umerical sequece is a ifiie se of umbers eumeraed by a posiive ieger ide i ascedig order of values of he ide I oher words, a sequece is a fucio f ( of a discree variable, whose domai cosiss of he se of all aural umbers The elemes of a sequece are called he erms The erm f ( (ha is, he -h erm is called he geeral erm or variable of he sequece The geeral erm is deoed by a lower case leer wih he subscrip : a, b,, ec The geeral erm pu io braces deoes a sequece: { a }, b } }, ec Graphically, a sequece ca be represeed by pois o he umber lie: { { Oe ca also use a wo-dimesioal char for preseig a sequece: A sequece is compleely deermied by is geeral erm If a sequece is give by a few firs erms, he we eed o fid he geeral erm Eamples: The geeral erm a deermies he sequece of eve umbers: { a },, 6, K,,K The geeral erm b q deermies a ifiie geomeric progressio wih he commo raio q: { b }, q, q, K, q,k { c },,,, K c ( { },, 6,,, 7, K,! 6

7 7 Sequeces ( { y },,,,,,,,, K y π 6 { z },,,,,,,, K z si 7 Le be he sum of he firs elemes of a sequece { a }: S S a k k The he se S, S, K, S, K is also a sequece { S } which is called he sequece of he parial sums of he sequece { a } Bouded Sequeces A sequece { } is said o be a upper-bouded sequece, if here eiss a fiie umber U such ha U for all aural umbers The umber U is said o be a upper boud of { } Ay oempy upper-bouded sequece has he leas upper boud A sequece { } is called a lower-bouded sequece if here eiss a fiie umber L such ha L for each aural umber The umber L is called a lower boud of { } Each oempy lower-bouded sequece has he greaes lower boud A sequece is called bouded, if here eis wo fiie umbers, L ad U, such ha L U for all erms of he sequece Oherwise, he sequece is ubouded Eamples: The sequece { },,,,, K,, K 9 6 is a bouded, sice < for all aural umbers The sequece 9 6 { },,,, K,, K is lower-bouded, sice ( for all aural umbers However, he sequece has o a fiie upper boud

8 Sequeces The sequece {( },, 8, 6, is ubouded, sice i has o fiie bouds K Moooe Icreasig Sequeces A sequece { } is called a moooe icreasig sequece, if for each aural umber A sequece { } is called a moooe decreasig sequece, if for each aural umber Eamples of moooe icreasig sequeces: { },,,,, K,, K 6 {!},, 6,, K,!, K Eamples of moooe decreasig sequeces: { },,,,, K,, K { },,,,, K,, K 8 6 Eamples of o-moooe sequeces: ( { },, π {si },,,, 8,,,,,, Sequeces ca also be classified o he basis of a behavior of is erms, if akes o sufficiely large values For isace, he variable icreases wihou ay boud wih icreasig I such a case, hey say ha he variable is ifiie large, ha is, i approaches ifiiy as eds o ifiiy Usig a symbolical form we wrie: as If a variable approaches zero as eds o ifiiy, he is said o be a ifiie small quaiy I his case we wrie: as A he same ime, some variables do o ed o ay well-defied umber as eds o ifiiy, eg, he erms of he sequece {( } oscillae bewee wo values, ad, K K

9 Illusraios Sequeces 9

10 Sequeces Limis of Numerical Sequeces Iuiive Defiiio of he Limi: The i of a sequece { } is a umber a such ha he erms remai arbirarily close o a whe is sufficiely large This saeme is wrie symbolically i ay of he followig form: a, a, a as I a mahemaical form, he saeme is sufficiely large meas sarig from some umber N ; he saeme he erms remai arbirarily close o a meas ha he absolue value of he differece bewee ad a is geig smaller ha ay arbirary small posiive umber δ Traslaig he above defiiio o he mahemaical laguage, we obai he followig Formal Defiiio of he Limi: Number a is called he i of a sequece { } if for ay arbirary small umber δ > here eiss a umber N such ha a < δ for each > N Geomerically, he iequaliy ierval ( a δ, a δ : a < δ ca be ierpreed as he ope I Calculus, he ierval ( a δ, a δ is usually amed dela eighborhood (or dela viciiy of he poi a I paricular, he dela eighborhood of zero is he ope ierval ( δ, δ : As a rule, we use he erm dela (or epsilo eighborhood, keepig i mid ha δ (or ε is a small posiive umber

11 Sequeces I erms of δ -eighborhood, he i of a sequece ca be defied by he followig wordig: Number a is he i of a sequece { } if ay arbirary small dela eighborhood of he poi a coais all erms of he sequece, sarig from a suiable erm I he figures below, he defiiio of he i is illusraed by he umber lie ad usig wo-dimesioal chars for some special cases The variable eds o a i a arbirary way The variable eds o a beig less ha a The variable eds o a beig greaer ha a

12 Sequeces If a sequece has a i a such ha a is a fiie umber, hey say ha he sequece coverges o he umber a, ad he sequece is called coverge Oherwise, he sequece is called diverge Eamples: The sequece {( } is diverge, sice i has o a i as eds o ifiiy The sequece { } is diverge, sice i approaches ifiiy as umber eds o ifiiy Prove ha he sequece { } coverges o he umber Iuiive Proof: If is a sufficiely large umber, he umber is much less ha ad i ca be egleced, ha resuls i as To prove his saeme rigorously, we have o show ha for ay arbirary small umber δ > here eiss a umber N such ha he codiio > N implies he iequaliy < δ Ideed, ( < δ < δ ( ( < δ 6 > δ > ( 6 δ Seig N ( we obai ha he iequaliy 6 δ > N implies > (, 6 δ ad hece, < δ, o maer how small posiive value of δ is chose

13 Ifiiesimal Sequeces A sequece { α } is called ifiiesimal, if i coverges o zero: α The Formal Defiiio is he followig Sequeces A sequece { α } is called a ifiiesimal sequece, if for ay arbirary small posiive umber δ here eiss a umber N such ha he iequaliy > N implies α < δ O he umber lie, he pois α of a ifiiesimal sequece come arbirarily close o he zero poi as icreases o ifiiy I meas ha he zero poi is he accumulaio poi for ay ifiiesimal sequece For > N (δ, o maer how small posiive δ is chose, all pois α remai i he dela eighborhood of zero I order o beer udersad he cocep of ifiiesimals, ry o imagie somehig divided io millios bis The, divided agai a obaied bi io millios bis Repeaig his procedure idefiiely may imes, we approach o a ifiiesimal bi Eamples: The sequece { },,,, K,, K is a ifiiesimal sequece, sice as Rigorous Proof: We eed o show ha for ay δ > here eiss a umber N such ha he iequaliy > N implies < δ

14 Sequeces If we se N, he he wo-sided iequaliy > N implies he δ δ desired iequaliy < δ for ay arbirarily small δ > I paricular, seig δ we obai < for all > Thus, he give dela eighborhood of he zero poi coais all erms of he sequece { } ecep for he firs hudred erms If δ he < for all >, ha is, all pois (wih > lie i he dela eighborhood of he zero poi I does o maer how may erms are beyod a dela viciiy of zero, if oly a fiie umber of erms does o belog o ha dela viciiy The variable is ifiiesimal as Proof: If we se N δ, he iequaliy > N implies > δ, ad so < δ for ay arbirary small δ > However, < Therefore, > N < δ < < δ Hece, The variable si is ifiiesimal as Proof: Give ay arbirary small δ >, > arcsiδ < arcsiδ si < δ If we se N, he arcsiδ iequaliy > N implies si < δ

15 Sequeces The cocep of ifiiesimals gives a more coveie ierpreaio of he i of a sequece By he defiiio, he saeme a meas ha a < δ for > N, ad hece he differece ( a is a ifiiesimal variable Therefore, we arrive a he followig helpful rule: Number a is he i of a sequece { }, if he geeral erm of he sequece ca be epressed i he form a α, where α is he geeral erm of a ifiiesimal sequece Eample: Evaluae he i of he sequece { } as Eplaaio: Number (i he deomiaor ca be egleced as The, which meas To give a Formal Proof, we should epress he geeral erm of he sequece as he sum of a cosa erm ad a ifiiesimal variable: ( The epressio ( is a ifiiesimal variable, sice as Therefore, he cosa erm is he i of he sequece,

16 Sequeces Ifiie Large Sequeces A sequece { } is called ifiie large (or diverge, if approaches ifiiy as eds o ifiiy The formal defiiio is he followig: Noaios: A sequece { } is called a ifiie large sequece, if for ay arbirary large umber > here eiss a umber N such ha > for each > N or,, as If he erms of a ifiie large sequece { } are, respecively, posiive or egaive a leas sarig from a sufficiely large umber N, we use he followig oaios: or Noe ha a - eighborhood of a ifiie poi icludes eiher oe of he semi-ifiie iervals, (, ad (,, or boh Terms belog o a - eighborhood of a ifiie poi, if heir absolue values are greaer ha ay arbirary large umber > 6

17 Sequeces Eample: The sequece { } is a ifiie large sequece sice as Rigorous Proof: We eed o show ha for ay arbirary large umber > > N Noe ha here eiss a aural umber N such ha > 7 > wheever If we se N, he he wo-sided iequaliy > N implies >, ad hece > > for ay arbirary large umber >

18 Sequeces Properies of Ifiiesimal Sequeces Propery If { α } is a ifiiesimal sequece ad { b } is a bouded sequece, he α } is a ifiiesimal sequece { b Eplaaio: Firs, accordig o he defiiio, all erms of he bouded sequece { b } are resriced by a fiie umber M, b < M Secod, a ifiiesimal variable α approaches zero as Hece, α b α b < α M M as Rigorous Proof: Sice { α } is a ifiiesimal sequece, he for ay arbirary small δ >, he posiive umber δ M correspods o a suiable δ umber N such ha α < for each > N Therefore, M δ α b α b < M δ M wheever > N, which required o be proved Propery If { α } ad { β } are ifiiesimal sequeces, he { α β} is also a ifiiesimal sequece Eplaaio: α β α β as Rigorous Proof: For ay δ >, he umber δ correspods o aural umbers N ad N such ha δ α < for each > N ad δ β < for each > N If a umber N is o less ha each of he umbers N ad N, he δ δ α β α β < δ for ay arbirary small δ > wheever > N Corollary: The sum of ay fiie umber of ifiiesimal variables is a ifiiesimal variable 8

19 Sequeces Eplaaio: The idea of a proof is show i he drawig below The sum of ay wo ifiiesimals ca be represeed by oe ifiiesimal The he sum of wo obaied ifiiesimal gives also a ifiiesimal, ec Rigorous Proof: Le us apply he mahemaical iducio priciple Se (wih,,, be he saeme The sum of ifiiesimals is a ifiiesimal Iducio basis: By propery, he saeme S is rue for Iducio hypohesis: Assume ha he saeme S holds rue for some ieger Iducio sep: By he hypohesis, he sum of ifiiesimals is a ifiiesimal, ad so he sum of ( ifiiesimals ca be cosidered as he sum cosisig of wo ifiiesimal iems However, he sum of wo ifiiesimals is a ifiiesimal by Propery Therefore, he saeme S implies he saeme S, ad hece S is rue for ay ieger S Propery If { α } is a ifiiesimal sequece he { } α ad vice versa is a ifiie large sequece Eplaaio: To verify he proposiio, divide umber by,,, ad so o The divide umber by,,, ad so o Compare he resuls Rigorous Proof: Le { α } be a ifiiesimal sequece The for ay > here eiss a umber N such ha < α, which implies > α for each > N Hece, { } is a ifiie large sequece α Likewise, if { α } is a ifiie large sequece, he ay δ > correspods o a umber N such ha α >, ad so δ δ α < wheever > N Hece, { } is a ifiiesimal sequece α 9

20 Sequeces Properies of Limis of Sequeces Propery ( c c Proof: Le a ha meas a α, where α is a ifiiesimal The for ay umber c, c c( a α c a cα Sice cα is a ifiiesimal, ( c c a, which required o be proved Propery If here eis fiie is of sequeces { } ad { y } he ( y y Proof: Le a ad y b ha meas a α, ad y b β, where α ad β are ifiiesimals The y ( a b ( α β By he properies of ifiiesimals, he sum ( α β is a ifiiesimal Therefore, ( y a b Propery If here eis fiie is of sequeces { } ad { y } he ( y y Proof: Likewise, he saemes a ad y b imply y ( a α ( b β a b ( bα a β α β I view of he properies of ifiiesimals, he variable ( bα a β α β is a ifiiesimal Therefore, ( y a b ( ( y

21 Sequeces Propery If here eis fiie is of sequeces { } ad { y }, ad y he y y Proof: Assume ha a ad y b To prove he propery, we have o represe he quoie i he form y a Ifiiesimal y b Usig simple rasformaios we obai y a α b β a b a b a α ( b β a b a b bα a b a β b( b β By he properies of ifiiesimals, bα a β b( b β b as Therefore, a y b y a b bα a β b( b β Propery p If here eis fiie is of sequeces { } ad { } he Eplaaio: Le a The where α is a ifiiesimal Therefore, p p ( ( a α a p p ( a α, a p p p α ( a a p p ( p

22 Sequeces Classificaio of Ifiiesimal Sequeces α Le λ, where α ad β are ifiiesimal variables as β If < λ <, he α ad β are called ifiiesimals of he same order of smalless α I paricular, if he α ad β are called equivale β ifiiesimals: α ~ β I ha case, hey say ha he ifiiesimals are equal asympoically If λ he α is called a ifiiesimal of a higher order of smalless wih respec o β, while β is a ifiiesimal of a lower order of smalless wih respec o α If λ he β is a ifiiesimal of a higher order of smalless wih respec o α, while α is a ifiiesimal of a lower order of smalless wih respec o β α If < <, he α k is called a ifiiesimal of he k-h ( β order of smalless wih respec o β Eamples: Ifiiesimals, sice ad ( Ifiiesimals ad are equal asympoically as have he same order as ( ( ha is a fiie umber Show ha ( respec o as, sice is a ifiiesimal of he secod order wih

23 Sequeces 6 Compariso Bewee Ifiiesimal Sequeces Le α ad β be wo equivale ifiiesimals The β α γ, where γ is a ifiiesimal of a higher order of smalless wih respec o boh α ad β Proof: By he defiiio, if α ~ β he β α as, ad so β α β ( ( α α Therefore, he differece ( β α is a ifiiesimal of a higher order of smalless wih respec o he give ifiiesimals If β is a ifiiesimal of a higher order of smalless wih respec o α, he α β ~ α I meas ha β is a egligible quaiy wih respec o α as Proof: By he hypohesis, β α as The α β β α β α ( α ~ ( α α α α Le α ad β be wo ifiiesimals of he same order α If λ he α ad λ β are equivale ifiiesimals: β α ~ λ β I ha case, he ifiiesimals are said o be proporioal asympoically α α Proof: λ as λβ λ β λ Eample: Sice ad are wo equivale ifiiesimals, he heir sum is a ifiiesimal of he same order of smalless: ~ However, heir differece is a ifiiesimal of he secod order wih respec o he give ifiiesimals: (

24 Sequeces 7 Classificaio of Ifiie Large Sequeces α Le λ, where α ad β are ifiie large variables as β If < λ < he α ad β are called ifiie large variables of he same icreasig order α I paricular, if he α ad β are called equivale ifiie β large variables: α ~ β I ha case, ifiie large variables are said o be equal asympoically If λ he α is called a ifiie large variable of a higher order of icrease wih respec o β, while β is a ifiie large variable of a lower icreasig order wih respec o α If λ he α is a ifiie large variable of lower order wih respec o β, while β is a ifiie large variable of a higher order wih respec o α α If < < he α k is called a ifiie large variable of ( β he k-h order of icrease wih respec o β Eamples: Two ifiie large variables, ad (, are equal asympoically as, sice Boh ifiie large variables, ad (, have he same icreasig order as, sice ( ha is a fiie umber The variable ( is a ifiie large variable of he hird order wih respec o sice (, ad is a fiie umber

25 Sequeces 8 Compariso bewee Ifiie Large Sequeces Le α ad β be wo equivale ifiie large variables The β α γ, where γ is a ifiiesimal of a lower icreasig order wih respec o boh α ad β Proof: By he defiiio, if α ~ β he β α as, ad so β α β ( ( α α Therefore, he differece ( β α is a ifiie large variable of a lower order of icrease wih respec o he give variables α β β Proof: α ~ β α α If β is a ifiie large variable of a lower icreasig order wih respec o α, he α β ~ α I meas ha β is a egligible quaiy wih respec o α as β Proof: By he hypohesis, as The α α β β α β ~ α α α Le α ad β be wo ifiie large variables of he same order α If λ he α ad λ β are equivale ifiie large variables: β α ~ λ β I ha case, he ifiie large variables are said o be proporioal asympoically α α Proof: λ as λβ λ β λ Eamples: I he epressio ( 7, he quaiy ( 7 is egligible wih respec o, sice 7 7 (

26 Sequeces 6 Two variables, ad, are equivale ifiie large variables of he secod order wih respec o Their sum is a ifiie large variable of he same icreasig order: ~ However, he differece bewee he give variables is a ifiie large variable of he firs icreasig order wih respec o : ~ ( ( To fid he i of he epressio 8 6 9, oe ha each of he variables, ad, is equivale o as Therefore, 9 ( 8 6 ( Likewise, ad Therefore, ~ ( ~ ( To fid he i of he epressio ( havig a ideermiae form ( as, muliply ad divide he differece ( by he sum ( o ge he differece bewee wo squares: ( ( ( Sice ~ ( ad ~ (, we obai (

27 9 Theorems of Sequeces Sequeces Theorem Each moooe icreasig upper-bouded sequece has a fiie i The below drawig illusraes he heorem Proof: Le a be he leas upper boud of he sequece { } I meas ha i all he erms of { } saisfy he iequaliy a ; ii for ay arbirary small posiive δ, he umber ( a δ is o a upper boud of he sequece Therefore, here eiss a erm N, which is greaer ha ( a δ : a δ < N However, { } is a moooe icreasig sequece, ad so N N N K Thus, all he successors saisfy jus he same iequaliies, comig arbirary close o he boud a: a δ < a for each N Hece, a Theorem Each moooe decreasig lower-bouded sequece has a fiie i The heorem ca be proved i a similar way Proof: Le a be he greaes lower boud a sequece { } I meas ha i all he erms of { } saisfy he iequaliy a ; ii for ay arbirary small posiive δ, he umber ( a δ is o a lower boud of he sequece The here eiss a erm N such ha a N < a δ Sice he sequece is moooe decreasig, all he successors remai bewee he boud a ad erm N : a N < a δ for each N, as i is show i he drawig below Hece, a 7

28 Sequeces Theorem A moooe icreasig sequece is diverge, if i has o a upper boud Proof: Le be a arbirary large umber Sice is o a upper boud of he sequece { }, here eiss a erm N, which is greaer ha However, { } is a moooe icreasig sequece, ad so each successor is also greaer ha Thus, for ay arbirary large umber here eiss he correspodig umber N such ha > wheever > N Hece, he heorem Theorem A moooe decreasig sequece is diverge, if i has o a lower boud Proof: By he argumes used i he proof of Theorem, we coclude ha for ay posiive umber here eiss he umber N such ha <, ad so > wheever > N Hece, he sequece diverges Number e Theorem: The sequece {( } is a moooe icreasig bouded sequece; has he fiie i such ha < ( < Tha i is deoed by he symbol e: e ( The umber e is a irraioal umber, e 788 8

29 Sequeces Proof: Firs, we prove ha { a } {( } is a moooe icreasig sequece By he Biomial Theorem (see [-], for eample: ( ( ( ( y y y y L!! ( ( L y! Seig ad y ad makig simple rasformaios we obai: ( ( (! a ( K!!! ( (!! ( ( ( (!! ( ( L K! K ( ( L(! a Subsiuig ( for we ge a similar epressio for he e erm : a (! ( ( K! ( ( L(! ( ( L( ( (! Now le us compare he epressios for ad erm by erm a a a a Firs of all, oe ha i boh sums (i he epressios for ad all he erms are posiive, ad he umber of he erms icreases wih icreasig Sarig from he secod erm, each erm of he sum a is greaer ha he correspodig erm of he sum for a : ( < (, ( ( < ( (, ad so o Therefore, iequaliy a > a proves ha { a } {( } is a moooe icreasig sequece 9

30 Sequeces Now le us prove ha { a } {( } is a bouded sequece The firs erm of a moooe icreasig sequece is he greaes lower boud of he sequece Sice a (, we ge he iequaliy ( which is valid for all aural umbers ha proves he eisece of a lower boud The eisece of a upper boud ca be proved by he followig simple esimaios Firs, a ( ( ( K ( ( L(!!! k sice < <!! K! for all aural umbers k ad Secod, < for all ieger k k >, ad so k! a < K The epressio o he righ side of his iequaliy icludes he sum of erms of he geomeric progressio wih he commo raio ha ca be easily calculaed []: ( K < Thus, { a } is a upper-bouded sequece, a ( < for each aural umber The sequece {( } saisfies he codiios of Theorem, ad so i has a fiie i deoed by he symbol e, ( e, < e <

31 Graphic Illusraio Sequeces Numerical illusraio ( e

32 Fucios FUNCTIONS Elemeary Fucios: Shor Review Power Fucios The domai of he power fucio y is he se of all real umbers ecep for <, if ( k,, K, ad, if < k The rage of he power fucio y depeds o he ide of he power If is a eve umber he he rage coais oly o-egaive real umbers For odd umbers, he rage is he se of all real umbers If he y is a liear fucio, whose graph is a sraigh lie passig hrough he origi Domai: The se of all real umbers Rage: The se of all real umbers Symmery: A odd fucio, y y( y( If he y is a quadraic fucio, whose graph is a parabola wih he vere a he origi Domai: The se of all real umbers Rage: The se of all o-egaive real umbers Symmery: A eve fucio, y ( y( If he y is a cubic fucio, whose graph passes hrough he origi Domai: The se of all real umbers Rage: The se of all real umbers Symmery: A odd fucio

33 Fucios If he he equaio y describes he hyperbola Domai: The se of all posiive ad egaive real umbers Rage: The se of all posiive ad egaive real umbers Symmery: A odd fucio If he y is he iverse fucio of y provided ha Domai: The se of all o-egaive real umbers Rage: The se of all o-egaive real umbers If he y is he iverse fucio of y Domai: The se of all real umbers Rage: The se of all real umbers Symmery: A odd fucio If he y Domai: The se of all real umbers Rage: The se of all posiive real umbers Symmery: A eve fucio

34 Fucios Epoeial Fucios Requiremes: a > ad a Domai: The se of all real umbers Rage: The se of all posiive real umbers Properies: If a > he y a is a moooe icreasig fucio, ha is, > a > a ; y a graphs of he fucio y a eds o he -ais asympoically as, ad eds o ifiiy as If < a < he y a is a moooe decreasig fucio, ha is, > a < a ; graphs of he fucio y a eds o he -ais asympoically as, ad eds o ifiiy as Graphs: Basic Formulas: a a a a a a a a a ( a a The reader ca fid more deail discussio of he properies of elemeary fucios, for eample, i [-]

35 Logarihmic Fucios y log Noe: a > ad a Domai: The se of all posiive umbers Rage: The se of all real umbers a Fucios The logarihmic fucio is defied as he iverse of he epoeial fucio: y y loga a Properies: If a > he y loga is a moooe icreasig fucio, ha is, > loga > loga ; log a ad graphs of he fucio y loga eds o he y- ais asympoically as ; log as a If < a < he log a is a moooe decreasig fucio, ha is, > loga < loga ; log a ad graphs of he fucio y loga eds o he y- ais asympoically as ; log as Graphs: a Sice y a ad y loga are iverse fucios of each oher, heir graphs look as he mirror images of each oher across he bisecor of he firs ad hird quadras, see he figures below

36 Fucios Basic Formulas: log a log a a loga loga loga ( loga loga loga log log log a log log a log The fucio log is referred o as log ( lg i Russia books The fucio loge is deoed by l ad i is called he aural logarihm 6 a c c a a

37 Trigoomeric Fucios Sie Fucio y si Cosie Fucio y cos Fucios The reader ca fid more deail discussio of he properies of rigoomeric fucios, for eample, i [-] Domais: The se of all real umbers Rages: si, cos Properies: si ad cos are periodic fucios wih period π : si( π si, cos( π cos ; si is a odd fucio: si( si ; cos is a eve fucio: cos( cos Graphs: 7

38 Fucios Basic Formulas: Addiio Formulas for Sies ad Cosies si( α ± β siα cos β ± si β cosα cos( α ± β cosα cos β m siα si β Double-Agle Formulas for Sies ad Cosies si α siα cosα cosα cos α si α Half-Agle Formulas for Sies ad Cosies α si cosα α cos cosα Relaioships bewee Sies ad Cosies cos α si α π π siα cos( α cos( α π π cosα si( α si( α Oher Formulas α ± β α m β siα ± si β si cos α β α β cosα cos β cos cos α β α β cosα cos β si si siα si β (cos( α β cos( α β cosα cos β (cos( α β cos( α β siα cos β (si( α β si( α β 8

39 Tage Fucio y a Fucios Domai: The se of all real umbers ecep for ieger Rage: The se of all real umbers Properies: a is a periodic fucio wih period π : a( π a ; a is a odd fucio: a( a Coage Fucio y co π π, where is ay Domai: The se of real umbers ecep for ieger Rage: The se of all real umbers Properies: co is a periodic fucio wih period π : co( π co ; co is a odd fucio: co( co π, where is ay Graphs: 9

40 Fucios Relaioships bewee Trigoomeric Fucios si a cos cos co si a co a cos co si π a co( π co a( Addiio Formulas for Tages ad Coages aα ± a β a( α ± β m aα a β coα co β m co( α ± β coα ± coβ Double-Agle Formulas for Tages ad Coages aα a α a α co α co α coα Half-Agle Formulas for Tages ad Coages α cosα siα a siα cosα α cosα siα co siα cosα

41 Fucios Oher Formulas si( α ± β aα ± a β cosα cos β si( β ± α coα ± co β siα si β a si a a cos a a a a Values of Trigoomeric Fucios for Special Agles: Degrees Agle Radias si cos a co udefied π 6 π 6 π 9 π udefied

42 Fucios Iverse Trigoomeric Fucios Iverse Sie Fucio is referred as y arcsi or as y si Domai: Rage: π π arcsi Properies: arcsi is a moooe icreasig fucio; si (si ; si(si The soluio se of he equaio si a : ( arcsia π, (, ±, ±, K Iverse Cosie Fucio is referred as y arccos or as y cos Domai: Rage: arccos π Properies: arccos is a moooe decreasig fucio; cos (cos ; cos(cos The soluio se of he equaio cos a : ± arccos a π, (, ±, ±, K Graphs:

43 Fucios Iverse Tage Fucio y arca (or y a π π Domai: The se of all real umbers Rage: < arca < Properies: arca is a moooe icreasig fucio; a (a ; a(a The soluio se of he equaio a a : arca a π, (, ±, ±, K Iverse Coage Fucio y co Domai: The se of all real umbers Rage: < co < π Properies: co is a moooe decreasig fucio; co (co ; co(co The soluio se of he equaio co a : co a π, (, ±, ±, K Graphs:

44 Fucios 6 Hyperbolic Fucios The hyperbolic sie sih is defied by he followig formula: e e sih Domai: The se of all real umbers Rage: The se of all real umbers The hyperbolic sie is a odd fucio because e e e e sih( sih The hyperbolic cosie cosh is defied as e e cosh Domai: The se of all real umbers Rage: The se of all o-egaive real umbers The hyperbolic cosie is a eve fucio, sice e e cosh( cosh

45 Fucios The hyperbolic age ah is defied as he raio bewee sih ad cosh : si e e ah cosh e e Domai: The se of all real umbers Rage: The se of real umbers < The hyperbolic age is a odd fucio due o he symmery properies of sih ad cosh The hyperbolic coage coh is he raio of cosh o sih : cos e e coh sih ah e e Domai: The se of all real umbers ecep for Rage: The se of all real umbers The hyperbolic coage is a odd fucio due o he symmery properies of sih ad cosh

46 Fucios Basic Formulas: Addiio Formulas for sih ad cosh sih( α ± β sihα cosh β ± sih β coshα cosh( α ± β coshα cosh β ± sihα sih β Double-Agle Formulas for sih ad cosh sih α sihα coshα cosh α cosh α sih α Half-Agle Formulas for sih ad cosh α sih coshα α cosh coshα Oher Formulas α ± β α m β sihα ± sih β sih cosh α β α β coshα cosh β cosh cosh α β α β coshα cosh β sih sih sihα sih β (cosh( α β cosh( α β coshα cosh β (cosh( α β cosh( α β sihα cosh β (sih( α β sih( α β Relaioships bewee Hyperbolic Fucios cosh α sih α, sih cosh ah, coh, ah, cosh sih coh ah, coh cosh sih 6

47 Limis of Fucios Preiary Discussio Fucios Le f ( ad he values of he variable belog o a small viciiy of he poi The i looks like evide ha he values of he fucio f ( lie i a small viciiy of, ha is, as I his eample we ca direcly subsiue o ge he i value of f ( as However, if a fucio is o defied a some poi a, we eed o use aoher way of lookig o fid he i value of f ( as a Someimes, similar problems ca be solved algebraically, for isace, a ( a( a f ( a a, as a a a We see ha f ( approaches a as eds o a Therefore, by he supplemeary codiio a, if a f ( a, if a a he domai of f ( ca be eeded o iclude he poi a a Pracically, we have foud he i of he give fucio as a I may oher cases he evaluaio of he is is more complicaed The above eample shows ha i is possible o operae wih epressios of he form by he i process Some oher ideermiae forms ca be reduced o he form by algebraic rasformaios For eample, if f ( ad g ( as a, he he fracio f g is a ideermiae form, which ca be reduced o he form by dividig f g boh he umeraor ad deomiaor by he produc ( f g :, g f where g ad f as a Usig is oe ca also ivesigae he asympoic behavior of fucios a ifiiy The comprehesio of is creaes he ecessary prerequisies for udersadig all oher coceps i Calculus 7

48 Fucios Basic Cocepios ad Defiiios Here we will give differe formulaios of is i order o demosrae he uiy of various approaches o his cocep Iuiive argumes will be combied wih rigorous proofs of proposiios Iuiive Defiiio: The i of a fucio f ( is a umber A such ha he values of f ( remai arbirarily close o A whe he idepede variable is sufficiely close o a specified poi a: f ( A a Oe ca also say ha he values of he fucio f ( approach he umber A as he variable eds o he poi a Oher Noaio: f ( A as a The above defiiio gives he geeral idea of is I ca be easily raslaed io he rigorous mahemaical laguage The words he values of a give fucio f ( remais arbirarily close o A mea ha f ( A is less ha ay umber ε >, o maer how small ε is chose The oly hig ha maers is how he fucio is defied i a small eighborhood of he i poi By a value of ε we se a accepable deviaio of f ( from he i value A, ha is, ε meas a variaio of f ( from A, which ca be disregarded The boud values A ε ad A ε deermie he correspodig ierval ( a δ, a δ of he values of he idepede variable aroud is i poi a 8

49 Fucios The above drawig illusraes ha for ay pois i he ierval a δ < < a δ, ( he correspodig values of f ( lie i he epsilo viciiy of he poi A, A ε < f ( < A ε ( Seig δ mi{ δ, δ}, we ca chage iequaliy ( by he iequaliy a δ < < a δ ( If codiio ( implies iequaliy (, he iequaliy ( implies iequaliy ( eve more I is more coveie o operae wih a symmeric dela eighborhood of he poi a, ad ohig more i his chage Formally, he i of a fucio is defied as follows: Le a fucio f ( be defied i some eighborhood of a poi a, icludig or ecludig a A umber A is called he i of f ( as eds o a, if for ay arbirary small umber ε > here eiss he correspodig umber δ δ ( ε > such ha he iequaliy a < δ implies f ( A < ε The iequaliy a < δ epresses he codiio ha values of he variable are i a immediae viciiy of he i poi a If a is a ifiie poi he ay eighborhood of a cosiss of sufficiely large values of, ad so i is ecessary o modify he above defiiio for case of If, he i of a fucio is defied by he followig wordig: Number A is called he i of f ( as, if for ay arbirary small umber ε > here eiss he correspodig umber ( ε > such ha he iequaliy > implies f ( A < ε There are wo special cases of grea imporace: f ( as a, ad f ( as a I he firs case, he i of he fucio equals zero, f (, ad ( a f is called a ifiiesimal fucio 9

50 Fucios If f ( A as eds o a, he he differece bewee f ( ad is i value A approaches zero as a I meas ha f ( A α( is a ifiiesimal fucio as a Therefore, if a umber A is he i of a fucio f ( as a, he f ( ca be epressed as f ( A α(, where α ( is a ifiiesimal fucio as eds o a Thus, we have obaied he followig helpful rule of fidig he i of a fucio: f ( A ifiiesimal as a f ( A a Eamples of ifiiesimal fucios: as 8 as si as si as π e as as l as l( as I he secod case, he saeme mahemaical wordig: f ( as a has he followig If for ay arbirary large umber E > here eiss he correspodig umber δ δ ( E > such ha he iequaliy a < δ implies f ( > E, he he fucio f ( has a ifiie i as eds o he poi a If f ( as a, he fucio is called a ifiie large fucio ha is wrie symbolically as f ( a or f ( as a The symbolical oaios f ( ad f ( a a mea ha ifiie fucio f ( is posiive defied or egaive defied, respecively, a leas i some sufficiely small viciiy of he poi a

51 Fucios Eamples of ifiie large fucios: as as e as e as a ± as π co as l as l as Illusraios

52 Fucios

53 Oe-Sided Limis Fucios Now suppose ha f ( A as a provided ha belogs o a righsided eighborhood of he poi a ( > a The he umber A is called he righ-sided i of f (, A f ( a Oe ca also use he followig symbolic form o epress his saeme: f ( A as a or simply f ( a The lef-sided i has a similar meaig If f ( A as a (ha is, < a, he A is he lef-sided i of f ( : A f ( f ( a a If eds o zero beig less ha zero, we whie, while he direcio of approachig o zero from he side of posiive values is deoed by he symbolical form I erms of ε δ, oe-sided is are defied as follows: The umber A is called he righ-sided i of f ( as eds o a, if for ay arbirary small ε > here eiss a posiive umber δ δ (ε such ha he iequaliy a < < a δ implies f ( A < ε Likewise, if f ( A < ε wheever a δ < < a, he A is he lefsided i of f ( as eds o a By he defiiio of he i, f ( A as a, o maer wha sequece of values of, covergig o a, is chose Therefore, he followig heorem holds rue: f ( has he i a he poi a if ad oly if here eis oe-sided is as a, which are equal o each oher: f ( f ( A f ( A a a Eample: Fid he i of he fucio a f ( a he poi Soluio: Cosider oe-sided is of f ( as ± : f ( ad f ( They differ from each oher, ad so f ( has o a i a

54 Fucios Propery : Properies of Ifiiesimal Fucios Le f ( be a fucio bouded a leas i some eighborhood of a poi a ad α ( be a ifiiesimal fucio as eds o a The he produc f ( α ( is a ifiiesimal fucio Eplaaio: The absolue values of he bouded fucio f ( are resriced by a fiie posiive umber M, f ( < M, for ay i some eighborhood of he poi a Sice α ( is a ifiiesimal fucio as eds o a, he f ( α ( < M α( M Rigorous Proof: Sice f ( is a fucio bouded i some eighborhood of he poi a, he here eiss a fiie posiive umber M such ha f ( < M ( wheever a < δ ( Sice α ( a ifiiesimal fucio i he same eighborhood of he poi a, ay posiive umber ε M correspods o a posiive umber δ such ha he iequaliy a < δ (6 implies ε α ( < (7 M Le us se δ mi{ δ, δ} The codiio a < δ implies iequaliies ( ad (6, ha resuls i iequaliies ( ad (7 Therefore, for ay arbirary small umber ε, we obai ha ε α ( f ( α( f ( < M ε M wheever he values of are i he dela eighborhood of he poi a This proves ha α ( f ( is a ifiiesimal fucio Propery : The sum of wo ifiiesimal fucios is a ifiiesimal fucio Eplaaio: If α ( ad β ( as a, he α ( β ( as a Rigorous Proof: Le α ( ad β ( be ifiiesimal fucios as a

55 Fucios The for ay arbirary small posiive umber ε here eis he correspodig umbers δ > ad δ > such ha ε a < δ α ( < ad ε a < δ β ( < If δ mi{ δ, δ } he iequaliy a < δ implies boh a < δ ad a < δ, ad hece ε ε α ( β ( α( β ( < ε for ay arbirary small umber ε Corollary: The sum of ay fiie umber of ifiiesimal fucios is a ifiiesimal fucio Eplaaio: The sum of wo ifiiesimals is a ifiiesimal, he sum of which wih a hird ifiiesimal is also a ifiiesimal, ad so o The saeme ca be proved rigorously by he mahemaical iducio priciple jus i he same maer ha was used i a case of sequeces (See Chaper, p 8-9 Eample: si a l( is a ifiiesimal fucio as, sice each erm of he sum is a ifiiesimal fucio Propery : Properies of Limis of Fucios A cosa facor ca be ake ou he sig of he i, c f ( c f ( a a Proof: By he rule formulaed o page 7, f ( A f ( A α(, a where α ( is a ifiiesimal fucio as a Therefore, c f ( c A cα( c f ( c A Properies -: a

56 Fucios If here eis boh is, f ( ad g(, he here eis he is of he sum, produc ad quoie of he fucios: a a ( f ( ± g( f ( ± g( a a a ( f ( g( f ( g( a a a f ( f ( a (if g( a g( g( a a Le us prove, for eample, Propery The saemes f ( A ad g( B a a mea ha f ( A α( ad g( B β (, where α ( ad β ( are ifiiesimal fucios as a Therefore, f ( g( ( A α( ( B β ( AB ( Aβ ( Bα( α( β ( Sice Aβ ( Bα( α( β ( is also a ifiiesimal fucio as a, he f ( g( AB f ( g( a a Eamples a a Evaluae si Soluio: I order o evaluae a ideermiae form, we eed o selec ad cacel commo ifiiesimal facors i he umeraor ad deomiaor of he give epressio: a si si si cos cos By he properies of is, cos cos 6

57 Fucios Evaluae Soluio: Here we deal wih a ideermiae form Trasform he epressio uder he sig of he i: ( ( ( ( ( ( ( ( ( We have obaied he epressio of he form cosa, ad so Fid (a ideermiae form Soluio: Prese he fracio i facored form; he cacel he commo ifiiesimal facors: ( ( ( ( Evaluae 6 (a ideermiae form Soluio: Usig he idea of reducig he commo facors, we obai 7 ( ( ( ( 6 Fid (a ideermiae form Soluio: I order o evaluae he ideermiae form, divide he umeraor ad deomiaor ad he apply he properies of is: ( ( ( ( 7

58 Fucios 6 Evaluae 7 (a ideermiae form Soluio: Muliply he umeraor ad deomiaor by he sum 7 ( o complee he differece bewee wo squares The cacel he like ifiiesimal facors: ( ( 7 ( ( ( ( 9 7 ( 7 ( ( 7 ( 7 ( 7 7 Evaluae (a ideermiae form Soluio: Likewise, complee he differece bewee squares o selec ad cacel he commo ifiiesimal facors Noe ha ( ( ad ( ( Therefore, 8 Evaluae 8 (a ideermiae form Soluio: I view of he formula of he differece bewee wo cubes, ( ( ( 8 9 ( ( ( ( 8

59 Fucios ( ( 7 ( ( ( 7 ( ( ( ( ( ( 7 7 ( ( ( 7 ( ( ( ( ( ( ( ( ( ( ( ( 6 Classificaio of Ifiiesimal Fucios Ifiiesimal fucios ca be classified i he same maer ha was used i he correspodig secio devoed o he sequeces Two ifiiesimal fucios, α ( ad β (, are called he ifiiesimal fucios of he same order of smalless as eds o a, if heir raio has a fiie o-zero i: α( < < a β ( I ha case, ifiiesimal fucios α ( ad β ( are said o be proporioal o each oher i some viciiy of he poi a α( I paricular, if, ifiiesimal fucios α ( ad β ( are a β ( called equivale as eds o a ha is deoed symbolically as α ( ~ β ( 9

60 Fucios A ifiiesimal fucio α ( has a higher order of smalless wih respec o β ( as eds o a, if α( a β ( I his case, β ( is called a ifiiesimal fucio of a lower order of smalless wih respec o α ( A ifiiesimal fucio α ( is called a ifiiesimal of he -h order wih respec o β ( as eds o a, if α ( ad (β ( are ifiiesimal fucios of he same order: α( < < a ( β ( Eamples Ifiiesimal fucios α( ad β ( as have he same order, sice α( ( ( (, β ( ad is a fiie o-zero umber If he α ( ad β ( are 8 ifiiesimal fucios of he same order, sice heir raio eds o a fiie o-zero umber: α( ( β ( ( The i of he raio of he ifiiesimal fucios α ( si ad β ( a as equals zero: α( si si cos cos β ( a si Therefore, α ( is a ifiiesimal fucio of a higher order of smalless wih respec o β ( 6

61 Fucios Le α ( ad β ( Sice α, β ( ha is, he i is a fiie umber, he α ( is a ifiiesimal fucio of he hird order wih respec o β ( as Give wo ifiiesimal fucios α ( ad β ( as, α β Therefore, α ( ad β ( are equivale ifiiesimal fucios as 6 Epressios α ( ad β ( deermie 7 equivale ifiiesimal fucios as, sice he i of heir raio equals uiy: α( β ( Compariso Bewee Ifiiesimal Fucios Rule : Le α ( ad β ( be wo equivale ifiiesimal fucios as a The β ( α( γ (, where γ ( is a ifiiesimal fucio of a higher order of smalless as a β ( Proof: By he defiiio, if α ( ~ β ( he as a, ad so α( β ( α( β ( α( α( Therefore, he differece ( β ( α( is a ifiiesimal fucio of a higher order of smalless wih respec o he give ifiiesimals as a 6

62 Fucios Rule : If β ( is a ifiiesimal fucio of a higher order of smalless wih respec o α ( as a, he α ( β ( ~ α( I meas ha β ( is a egligible quaiy wih respec o α ( as a β ( Proof: By he hypohesis, as a The α( α( β ( β ( α ( β ( α( ( α( ~ ( α( α( α( α( Rule : If α ( ad β ( are ifiiesimal fucios of he same order ad α( < λ <, β ( he α ( ad λ β ( are equivale ifiiesimal fucios, α ( ~ λ β ( Eamples: Sice ~ 7 7 he (7 (8 (7 6 (8 ad ~, Ifiiesimal fucios ad have higher orders of smalless wih respec o as Therefore, ~, ha is, ( is a egligible quaiy wih respec o as eds o zero Sice a ad si are equivale ifiiesimal fucios as, he differece bewee hem is a ifiiesimal fucio of a higher order of smalless wih respec o each of hem Really, si a si si si ( cos cos cos ( cos ( cos si si cos cos ( cos si ( cos si cos si cos ( cos cos ( cos as Thus, a si is a ifiiesimal fucio of he hird order of smalless wih respec o boh fucios, si ad a, as 6 8 7

63 8 Classificaio of Ifiie Large Fucios Fucios Two ifiie large fucios, α ( ad β (, have he same icreasig order as a, if heir raio has a fiie ozero i, α( < < a β ( I paricular, if α( a β ( he α ( ad β ( are called equivale ifiie large fucios as a ha is deoed symbolically as α ( ~ β ( A ifiie large fucio α ( has a higher icreasig order wih respec o β ( as eds o a, if α( a β ( Correspodigly, β ( is a ifiie large fucio of a lower icreasig order wih respec o α ( Le α ( ad (β ( be wo ifiie large fucios of he same order, α( < < a ( β ( The α ( is called a ifiie large fucio of he -h order wih respec o β ( as eds o a 9 Compariso Bewee Ifiie Large Fucios Rule : Le α ( ad β ( be wo ifiie large fucios as a The β ( α( γ (, where γ ( is a ifiie large fucio of a lower icreasig order as a Rule : The differece bewee wo equivale ifiie large fucios is a quaiy of a lower icreasig order: α( β ( β ( α ( ~ β ( α( α( Rule : If α ( is a ifiie large fucio of a higher icreasig order wih respec o β ( as a he he sum α ( β ( is a ifiie large fucio equivale o α (: 6

64 Fucios α( β ( β ( α ( β ( ~ α( α( α( I his case, β ( is said o be a egligible quaiy wih respec o α ( as a If α ( ad β ( are ifiie large fucios of he same icreasig order α( as a ad < λ <, he α ( ad λ β ( are equivale a β ( ifiie large fucios, α ( ~ λ β ( I his case, hey say ha ifiie large fucios are proporioal asympoically as a Eamples Ifiie large fucios f ( ad g( as have he same icreasig order, sice f g is a fiie umber: g( ( f ( Moreover, he i equals, ad so f ( ad g ( are equivale ifiie large fucios as Oe ca see ha (i he umeraor is a egligible quaiy wih respec o as Geerally, if k < ad a is a fiie umber, he ay power k fucio a is a egligible quaiy wih respec o as For isace, 7 ~ ad 6 8 ~ 6 Hece, The ifiie large fucio 7 is equal asympoically o as, sice 7 is a egligible quaiy wih respec o Likewise, 8 ~, sice 8 is a egligible quaiy wih respec o Therefore, 7 8 6

65 Evaluae Soluio: Sice ad ~ ~ 6, as, he Fucios Boh fucios, f ( ad g (, are ifiie large fucios as Prove ha: f ( ~ g( ; boh fucios, f ( ad g (, are ifiie large fucios of he secod icreasig order wih respec o as ; he differece f ( g( is a quaiy of a lower icreasig order wih respec o he give fucios f ( ad g ( Soluio: f ( g( ( ad ( By he formula of differece bewee wo squares, If ad ( f f g he g( f g f g f f g g ~ ( ( ~ Hece, f g ~, ha is, he differece is a ifiie large fucio of he firs icreasig order wih respec o 6

66 Fucios The Mos Impora Limis Theorem si This saeme ca be also epressed as si ~ as si Proof: Noe ha is a eve fucio Therefore, we may cosider oly he case of posiive values of variable i a viciiy of zero Le be a ceral agle (i radias of he ui circle Compare he areas of he figures show i he drawigs below The area of he riagle OAB is S OAB si The area of he circular secor OAB is S OAB The area of he riagle OAC is S OAC a Evidely, si < < a for ay < < π 66

67 si Recall ha a ad perform simple algebraic rasformaios: cos si < < a < < si cos si > > cos si If he cos, ad hece Graphic Illusraios: Fucios Noe: If α ( is a ifiiesimal fucio as a, he siα ( ~ α( (as a, siα(, a α( idepedely of a ype of he fucio α ( ad value of a The oly hig ha maers is a smalless of α ( as eds o a For isace, si( 8 ~ ( 8 as, si ~ as Eamples: si 7 si Aoher Soluio: 7 7 si 7 7 si 7 7 si 7 si 67

68 Fucios a si si cos cos Therefore, a ~ as arcsi Evaluae Soluio: By chagig he variable si we obai arcsi ad as Therefore, arcsi si Thus, ifiiesimal fucio arcsi is equivale o as, arcsi ~ as arca Evaluae Soluio: Likewise, subsiuio as Therefore, arca cos Evaluae 68 arca implies a ad a arca ~ as, Soluio: By makig use of he half-agle ideiy he umeraor ca be epressed hrough a sie fucio: cos si I view of Theorem, si ~ Therefore, si ( cos I meas ha cos ~, or cos ~ as

69 6 Evaluae si cos π a Fucios si si cos Soluio: Usig he ideiy a we ge cos cos si cos si cos cos cos π a π si cos π 7 Evaluae cos8 cos Soluio: By he rigoomeric ideiy cos si, cos8 si ( 6 cos si cos 8 Evaluae Soluio: Sice cos si ad si ~, he cos si si 9 Evaluae a Soluio: If he si ~ ad a ~ Therefore, si a 6 Evaluae si Soluio: Sice is a ifiiesimal fucio as, he si ~ Evaluae, ad so si a 7 a π π Soluio: I order o evaluae a ideermiae form, se : 69

70 Fucios ad Sice π 7π a 7 a 7( a(7 co 7 a 7 π a a ( a π, he π Therefore, a 7 a π a a 7 7 cos Evaluae cos Soluio: Trasform he umeraor: ( cos ( cos ( cos cos cos cos si ( cos ~ cos cos Now rasform he deomiaor: The we ge Evaluae cos si cos cos a( π Soluio: Subsiuio implies 7 ~ a( π a( π π aπ Sice, he Therefore, a π ~ π ha resuls i a( π a( π π π 7

71 Calculus of Approimaios Fucios Here ad below we use he radia measure of agles uless he corary is allowed Theorem saes he approimaio formula si, which is valid for ay i some small viciiy of zero Oher rigoomeric fucios ca be epressed hrough he sie fucio For isace, cos si (, si a cos The below drawigs illusrae he error rage of he above approimaio cos ( formulas A measure of iaccuracy % is show i he cos addiioal widow We ca hardly ever see ay differeces bewee graphs of fucios y cos ad y for < 8 rad Fig Graphs of fucios y cos (upper curve ad y (lower curve 7

72 Fucios The raio of he cosie fucio o is polyomial approimaio is show i Fig Oe ca see ha he quadraic polyomial fis well he cosie fucio i a wide rage of values of I Fig, he graphs of he fucios y a ad y are preseed A measure of iaccuracy a % a is show i he addiioal widow Fig The graph of he fucios y a ad y 7

73 Cosider a few umerical eamples Approimaio formulas for f ( si cos a Approimae values of f ( π π si si π π si si 6 6 π π cos cos ( π π cos cos π π a a π π a a Fucios Eac values of f ( Theorem ( 7 e (e 788 Usig subsiuio ad he reurig o he symbol, we ca epress Theorem i he oher form: ( e (See deailed discussio of he heorem i Chaper, pp 8- Noe: If α ( is a ifiiesimal fucio as a For isace, α ( ( α ( e a si ( si e, e, he

74 Fucios The well-kow logarihmic ideiy b bl a a e ca be geeralized by he i process ha resuls i he followig Impora Rule: If α ( ad β ( are ifiiesimal fucios as a ( a Le f ( ad g ( as a The g( ( f (, he l( α ( β ( a β ( α( e (8 is a ideermiae form as a g( To apply Theorem, i is ecessary o reduce ( ( α ( f o he sadard form ( α(, where α ( is a ifiiesimal fucio as eds o ifiiy The geeral procedure of he reducig is he followig: f g [ ] where α f Thus, a ( ( f g g ( f g ( f ( ( f f [ α] α, g( ( f ( ( ( f ( g( ( f ( f ( a e g( ( f ( a ha is he give problem is reduced o evaluaio of g( ( f ( a, (9 Eamples: Evaluae Soluio: Sice ( e as, he e 7

75 Evaluae Fucios Soluio: Noe ha, where is a ideermiae form as, while he epressio ( eds o as Therefore, Evaluae Soluio: ( ( e Evaluae Soluio: ( ( ( ( Evaluae ( Soluio: ( ( ( ( ( 8 ( 8 e e e e e 8 8 ( ( ( e e 7

76 Fucios Theorem l( This saeme ca also be epressed i he form l( ~ as Proof: By he properies of logarihms, l( l( Recall ha ( e as However, if he quaiy ( approaches umber e, he is aural logarihm eds o l e, Graphic Illusraios: l( l e as 76

77 Noe: If α ( is a ifiiesimal fucio as a, he l( α( ~ α( as a, l( α( a α( For isace, l( ~ as, Eamples: l( si ~ si as, l( 77 l( Evaluae 7 Soluio: I view of he above oe, l( l( l( si Evaluae a Soluio: By Theorem, si ~ ad a ~ as I addiio, l( si ~ si ~ as Therefore, l( si a Evaluae l( 9 arcsi a Fucios Soluio: Firs, (9 arcsi is ifiiesimal fucio as, ad so l( 9 arcsi ~ (9 arcsi Secod, arcsi ~ is a egligible quaiy wih respec o 9 as, ad so (9 arcsi ~ 9 Third, a ~ as Fially, l( 9 arcsi 9 a