SECTION 87 Intrminat Forms an L Hôpital s Rul 567 Sction 87 Intrminat Forms an L Hôpital s Rul Rcogniz its that prouc intrminat forms Apply L Hôpital s Rul to valuat a it Intrminat Forms Rcall from Chaptrs an that th forms 00 an ar call intrminat bcaus thy o not guarant that a it ists, nor o thy inicat what th it is, if on os ist Whn you ncountr on of ths intrminat forms arlir in th tt, you attmpt to rwrit th prssion by using various algbraic tchniqus Intrminat Form 0 0 Limit Algbraic Tchniqu Divi numrator an nominator by Divi numrator an nominator by 8 7 6 5 y y = Th it as approachs 0 appars to b Figur 8 Occasionally, you can tn ths algbraic tchniqus to fin its of transcnntal functions For instanc, th it 0 proucs th intrminat form 00 Factoring an thn iviing proucs 0 0 0 Howvr, not all intrminat forms can b valuat by algbraic manipulation This is oftn tru whn both algbraic an transcnntal functions ar involv For instanc, th it 0 proucs th intrminat form 00 Rwriting th prssion to obtain 0 mrly proucs anothr intrminat form, Of cours, you coul us tchnology to stimat th it, as shown in th tabl an in Figur 8 From th tabl an th graph, th it appars to b (This it will b vrifi in Eampl ) 0 00 000 0 000 00 0 0865 8 980 998? 00 00 689
568 CHAPTER 8 Intgration Tchniqus, L Hôpital s Rul, an Impropr Intgrals GUILLAUME L HÔPITAL (66 70) L Hôpital s Rul is nam aftr th Frnch mathmatician Guillaum François Antoin L Hôpital L Hôpital is crit with writing th first tt on iffrntial calculus (in 696) in which th rul publicly appar It was rcntly iscovr that th rul an its proof wr writtn in a lttr from John Brnoulli to L Hôpital I acknowlg that I ow vry much to th bright mins of th Brnoulli brothrs I hav ma fr us of thir iscovris, sai L Hôpital MathBio L Hôpital s Rul To fin th it illustrat in Figur 8, you can us a thorm call L Hôpital s Rul This thorm stats that unr crtain conitions th it of th quotint fg is trmin by th it of th quotint of th rivativs f g To prov this thorm, you can us a mor gnral rsult call th Etn Man Valu Thorm THEOREM 8 Th Etn Man Valu Thorm If f an g ar iffrntiabl on an opn intrval a, b an continuous on a, b such that g 0 for any in a, b, thn thr ists a point c in a, b such that fc fb fa gc gb ga NOTE To s why this is call th Etn Man Valu Thorm, consir th spcial cas in which g For this cas, you obtain th stanar Man Valu Thorm as prsnt in Sction Th Etn Man Valu Thorm an L Hôpital s Rul ar both prov in Appni A THEOREM 8 L Hôpital s Rul Lt f an g b functions that ar iffrntiabl on an opn intrval a, b containing c, cpt possibly at c itslf Assum that g 0 for all in a, b, cpt possibly at c itslf If th it of fg as approachs c proucs th intrminat form 00, thn f f c g c g provi th it on th right ists (or is infinit) This rsult also applis if th it of fg as approachs c proucs any on of th intrminat forms,,, or FOR FURTHER INFORMATION To nhanc your unrstaning of th ncssity of th rstriction that g b nonzro for all in a, b, cpt possibly at c, s th articl Countrampls to L Hôpital s Rul by R P Boas in Th Amrican Mathmatical Monthly MathArticl NOTE Popl occasionally us L Hôpital s Rul incorrctly by applying th Quotint Rul to fg B sur you s that th rul involvs fg, not th rivativ of fg L Hôpital s Rul can also b appli to on-si its For instanc, if th it of fg as approachs c from th right proucs th intrminat form 00, thn f c g f c g provi th it ists (or is infinit)
SECTION 87 Intrminat Forms an L Hôpital s Rul 569 TECHNOLOGY Numrical an Graphical Approachs Us a numrical or a graphical approach to approimat ach it a 0 b 0 c 0 5 0 What pattrn o you obsrv? Dos an analytic approach hav an avantag for ths its? If so, plain your rasoning EXAMPLE Evaluat Solution 0 0 Intrminat Form 0/0 Bcaus irct substitution rsults in th intrminat form 00 you can apply L Hôpital s Rul as shown blow 0 0 0 0 0 0 0 Apply L Hôpital s Rul Diffrntiat numrator an nominator Evaluat th it Try It Eploration A Eploration B NOTE In writing th string of quations in Eampl, you actually o not know that th first it is qual to th scon until you hav shown that th scon it ists In othr wors, if th scon it ha not ist, it woul not hav bn prmissibl to apply L Hôpital s Rul Anothr form of L Hôpital s Rul stats that if th it of fg approachs (or ) proucs th intrminat form 00 or, thn f f g g provi th it on th right ists as EXAMPLE Intrminat Form / Evaluat ln NOTE Try graphing y ln an y in th sam viwing winow Which function grows fastr as approachs? How is this obsrvation rlat to Eampl? Solution Bcaus irct substitution rsults in th intrminat form, you can apply L Hôpital s Rul to obtain ln ln 0 Apply L Hôpital s Rul Diffrntiat numrator an nominator Evaluat th it Try It Eploration A Eploration B
570 CHAPTER 8 Intgration Tchniqus, L Hôpital s Rul, an Impropr Intgrals Occasionally it is ncssary to apply L Hôpital s Rul mor than onc to rmov an intrminat form, as shown in Eampl EXAMPLE Applying L Hôpital s Rul Mor Than Onc Evaluat Solution Bcaus irct substitution rsults in th intrminat form, you can apply L Hôpital s Rul This it yils th intrminat form, so you can apply L Hôpital s Rul again to obtain 0 Try It Eploration A In aition to th forms 00 an, thr ar othr intrminat forms such as 0,, 0, 0 0, an For ampl, consir th following four its that la to th intrminat form 0 0, 0,, Limit is Limit is Limit is 0 Limit is Bcaus ach it is iffrnt, it is clar that th form is intrminat in th sns that it os not trmin th valu (or vn th istnc) of th it Th following ampls inicat mthos for valuating ths forms Basically, you attmpt to convrt ach of ths forms to 00 or so that L Hôpital s Rul can b appli 0 EXAMPLE Intrminat Form 0 Evaluat Solution Bcaus irct substitution proucs th intrminat form 0, you shoul try to rwrit th it to fit th form 00 or In this cas, you can rwrit th it to fit th scon form Now, by L Hôpital s Rul, you hav 0 Try It Eploration A Eploration B
SECTION 87 Intrminat Forms an L Hôpital s Rul 57 If rwriting a it in on of th forms 00 or os not sm to work, try th othr form For instanc, in Eampl you can writ th it as which yils th intrminat form 00 As it happns, applying L Hôpital s Rul to this it proucs which also yils th intrminat form 00 Th intrminat forms, 0, an 0 0 aris from its of functions that hav variabl bass an variabl ponnts Whn you prviously ncountr this typ of function, you us logarithmic iffrntiation to fin th rivativ You can us a similar procur whn taking its, as shown in th nt ampl EXAMPLE 5 Intrminat Form 5 y = ( + ) 6 Evaluat Solution Bcaus irct substitution yils th intrminat form, you can proc as follows To bgin, assum that th it ists an is qual to y y Taking th natural logarithm of ach si proucs ln y ln Bcaus th natural logarithmic function is continuous, you can writ ln y ln Intrminat form 0 ln Intrminat form 00 L Hôpital s Rul Now, bcaus you hav shown that ln y, you can conclu that y an obtain Th it of as approachs infinity is Figur 85 You can us a graphing utility to confirm this rsult, as shown in Figur 85 Eitabl Graph Try It Eploration A
57 CHAPTER 8 Intgration Tchniqus, L Hôpital s Rul, an Impropr Intgrals L Hôpital s Rul can also b appli to on-si its, as monstrat in Eampls 6 an 7 EXAMPLE 6 Intrminat Form 0 0 Fin sin 0 Solution Bcaus irct substitution proucs th intrminat form 0 0, you can proc as shown blow To bgin, assum that th it ists an is qual to y y sin 0 ln y ln 0 sin 0 lnsin 0 lnsin lnsin 0 cot 0 0 tan 0 sc 0 Intrminat form Tak natural log of ach si Continuity Intrminat form 0 Intrminat form L Hôpital s Rul Intrminat form 00 L Hôpital s Rul Now, bcaus ln y 0, you can conclu that y 0, an it follows that 0 sin 0 0 Try It Eploration A Opn Eploration y = (sin ) TECHNOLOGY Whn valuating complicat its such as th on in Eampl 6, it is hlpful to chck th rasonablnss of th solution with a computr or with a graphing utility For instanc, th calculations in th following tabl an th graph in Figur 86 ar consistnt with th conclusion that sin approachs as approachs 0 from th right 0 0 00 000 0000 00000 sin 085 079 09550 099 0999 09999 Th it of sin is as approachs 0 from th right Figur 86 Us a computr algbra systm or graphing utility to stimat th following its: an cos 0 0 tan Thn s if you can vrify your stimats analytically
SECTION 87 Intrminat Forms an L Hôpital s Rul 57 EXAMPLE 7 Intrminat Form STUDY TIP In ach of th ampls prsnt in this sction, L Hôpital s Rul is us to fin a it that ists It can also b us to conclu that a it is infinit For instanc, try using L Hôpital s Rul to show that Evaluat ln Solution Bcaus irct substitution yils th intrminat form, you shoul try to rwrit th prssion to prouc a form to which you can apply L Hôpital s Rul In this cas, you can combin th two fractions to obtain ln ln ln Now, bcaus irct substitution proucs th intrminat form 00, you can apply L Hôpital s Rul to obtain ln This it also yils th intrminat form 00, so you can apply L Hôpital s Rul again to obtain ln ln ln ln ln ln Try It Eploration A Eploration B Th forms 00,,, 0, 0 0,, an hav bn intifi as intrminat Thr ar similar forms that you shoul rcogniz as trminat 0 0 0 Limit is positiv infinity Limit is ngativ infinity Limit is zro Limit is positiv infinity (You ar ask to vrify two of ths in Erciss 06 an 07) As a final commnt, rmmbr that L Hôpital s Rul can b appli only to quotints laing to th intrminat forms 00 an For instanc, th following application of L Hôpital s Rul is incorrct 0 0 Incorrct us of L Hôpital s Rul 0 Th rason this application is incorrct is that, vn though th it of th nominator is 0, th it of th numrator is, which mans that th hypothss of L Hôpital s Rul hav not bn satisfi
57 CHAPTER 8 Intgration Tchniqus, L Hôpital s Rul, an Impropr Intgrals Erciss for Sction 87 Th symbol Click on Click on inicats an rcis in which you ar instruct to us graphing tchnology or a symbolic computr algbra systm to viw th complt solution of th rcis to print an nlarg copy of th graph Numrical an Graphical Analysis In Erciss, complt th tabl an us th rsult to stimat th it Us a graphing utility to graph th function to support your rsult sin 5 0 sin f 0 f 5 00 f 6 f In Erciss 5 0, valuat th it (a) using tchniqus from Chaptrs an an (b) using L Hôpital s Rul 5 6 9 sin 7 8 0 9 5 0 5 0 0 0 0 00 00 0 0 000 000 In Erciss 6, valuat th it, using L Hôpital s Rul if ncssary (In Ercis 8, n is a positiv intgr) 0 5 ln 6 0 7 8 0 0 n 9 sin sin a 0 0 sin 0 sin b 0 0 000 00 0 000 00 0 0 0 0 5 0 5 arcsin arctan 0 5 6 7 8 9 0 cos sin ln ln 5 6 In Erciss 7 5, (a) scrib th typ of intrminat form (if any) that is obtain by irct substitution (b) Evaluat th it, using L Hôpital s Rul if ncssary (c) Us a graphing utility to graph th function an vrify th rsult in part (b) 7 8 ln 9 0 5 0 6 7 0 8 9 ln 50 0 cos 5 5 8 5 ln 0 5 0 In Erciss 55 58, us a graphing utility to (a) graph th function an (b) fin th rquir it (if it ists) 55 56 57 sin 0 ln 5 0 sin 5 58 cot 0 tan 0
SECTION 87 Intrminat Forms an L Hôpital s Rul 575 Writing About Concpts 59 List si iffrnt intrminat forms 60 Stat L Hôpital s Rul 6 Fin th iffrntiabl functions f an g that satisfy th spcifi conition such that f 0 an g 0 5 Eplain how you obtain your answrs (Not: Thr ar many corrct answrs) (a) (c) 5 5 f 0 g f g 5 6 Numrical Approach Complt th tabl to show that vntually ovrpowrs ln (b) 6 Fin iffrntiabl functions f an g such that f g f g 5 5 an f g 0 Eplain how you obtain your answrs (Not: Thr ar many corrct answrs) In Erciss 7 7, fin any asymptots an rlativ trma that may ist an us a graphing utility to graph th function (Hint: Som of th its rquir in fining asymptots hav bn foun in prcing rciss) 7 y, > 0 7 y, 7 y 7 Think About It In Erciss 75 78, L Hôpital s Rul is us incorrctly Dscrib th rror 75 76 77 78 0 0 sin 0 0 cos cos sin 0 cos 0 y ln > 0 ln 0 0 0 0 6 0 8 0 0 Analytical Approach In Erciss 79 an 80, (a) plain why L Hôpital s Rul cannot b us to fin th it, (b) fin th it analytically, an (c) us a graphing utility to graph th function an approimat th it from th graph Compar th rsult with that in part (b) 6 Numrical Approach Complt th tabl to show that vntually ovrpowrs 5 79 80 tan sc 5 Comparing Functions In Erciss 65 70, us L Hôpital s Rul to trmin th comparativ rats of incras of th functions f m, an whr n > 0, m > 0, an 5 65 66 ln 67 68 ln 69 n 70 5 0 0 0 0 50 00 m g n, h ln n ln m n Graphical Analysis In Erciss 8 an 8, graph f /g an f/g nar 0 What o you notic about ths ratios as 0? How os this illustrat L Hôpital s Rul? 8 f sin, g sin 8 f, g 8 Vlocity in a Rsisting Mium Th vlocity v of an objct falling through a rsisting mium such as air or watr is givn by v k kt v 0 kkt whr v 0 is th initial vlocity, t is th tim in scons, an k is th rsistanc constant of th mium Us L Hôpital s Rul to fin th formula for th vlocity of a falling boy in a vacuum by fiing v 0 an t an ltting k approach zro (Assum that th ownwar irction is positiv)
576 CHAPTER 8 Intgration Tchniqus, L Hôpital s Rul, an Impropr Intgrals 8 Compoun Intrst Th formula for th amount A in a savings account compoun n tims pr yar for t yars at an intrst rat r an an initial posit of P is givn by Us L Hôpital s Rul to show that th iting formula as th numbr of compounings pr yar bcoms infinit is givn by A P rt 85 Th Gamma Function Th Gamma Function n is fin in trms of th intgral of th function givn by f n, n > 0 Show that for any fi valu of n, th it of f as approachs infinity is zro 86 Tractri A prson movs from th origin along th positiv y-ais pulling a wight at th n of a -mtr rop (s figur) Initially, th wight is locat at th point, 0 (a) Show that th slop of th tangnt lin of th path of th wight is (b) Us th rsult of part (a) to fin th quation of th path of th wight Us a graphing utility to graph th path an compar it with th figur (c) Fin any vrtical asymptots of th graph in part (b) () Whn th prson has rach th point 0,, how far has th wight mov? In Erciss 87 90, apply th Etn Man Valu Thorm to th functions f an g on th givn intrval Fin all valus c in th intrval a, b such that 87 88 89 90 A P r n nt 0 8 6 y y fc fb fa gc gb ga f, Functions f, f sin, f ln, 6 8 Wight (, y) 0 g g cos g g Intrval 0,, 0,, Tru or Fals? In Erciss 9 9, trmin whthr th statmnt is tru or fals If it is fals, plain why or giv an ampl that shows it is fals 9 0 9 If y, thn 9 If p is a polynomial, thn p 0 9 If f, thn f g 0 g 95 Ara Fin th it, as approachs 0, of th ratio of th ara of th triangl to th total sha ara in th figur 96 In Sction, a gomtric argumnt (s figur) was us to prov that (a) Writ th ara of ABD in trms of (b) Writ th ara of th sha rgion in trms of (c) Writ th ratio R of th ara of ABD to that of th sha rgion () Fin Continuous Functions In Erciss 97 an 98, fin th valu of c that maks th function continuous at 0 sin, 0 97 f c, 0 98 (, cos ) (, cos ) 0 y π sin θ 0 D A B 0 R π C f, c, 99 Fin th valus of a an b such that n 0 y 0 0 0 00 Show that for any intgr n > 0 0 y f() = cos π π a cos b
SECTION 87 Intrminat Forms an L Hôpital s Rul 577 0 (a) Lt f b continuous Show that f h f h f h 0 h (b) Eplain th rsult of part (a) graphically y h 0 Lt f b continuous Show that f h f f h f h 0 h 0 Sktch th graph of g, 0 0, 0 an trmin g0 0 Us a graphing utility to graph f k k for k, 0, an 00 Thn valuat th it k k 0 k 05 Consir th it ln 0 (a) Dscrib th typ of intrminat form that is obtain by irct substitution (b) Evaluat th it (c) Us a graphing utility to vrify th rsult of part (b) FOR FURTHER INFORMATION For a gomtric approach to this rcis, s th articl A Gomtric Proof of ln 0 by John H Mathws in th Collg 0 Mathmatics Journal MathArticl f + h 08 Prov th following gnralization of th Man Valu Thorm If f is twic iffrntiabl on th clos intrval a, b, thn fb fa fab a b 09 Intrminat Forms Show that th intrminat forms 0 0, 0, an o not always hav a valu of by valuating ach it (a) ln ln 0 (b) ln ln (c) ln 0 0 Calculus History In L Hôpital s 696 calculus ttbook, h illustrat his rul using th it of th function f a a a a a as approachs a, a > 0 Fin this it Consir th function h sin (a) Us a graphing utility to graph th function Thn us th zoom an trac faturs to invstigat h (b) Fin h analytically by writing h sin (c) Can you us L Hôpital s Rul to fin h? your rasoning Evaluat a f tt b t Putnam Eam Challng a a whr a > 0, a Eplain This problm was compos by th Committ on th Putnam Priz Comptition Th Mathmatical Association of Amrica All rights rsrv 06 Prov that if f 0, f 0, an g, thn a a fg 0 a 07 Prov that if f 0, f 0, an a thn a fg a g,