0_005.qd //0 : PM Page 98 98 CHAPTER Applicaions of Differeniaion f() as Secion.5 f() = + f() as The i of f as approaches or is. Figure. Limis a Infini Deermine (finie) is a infini. Deermine he horizonal asmpoes, if an, of he graph of a funcion. Deermine infinie is a infini. Limis a Infini This secion discusses he end behavior of a funcion on an infinie Consider he graph of f inerval. as shown in Figure.. Graphicall, ou can see ha he values of f appear o approach as increases wihou bound or decreases wihou bound. You can come o he same conclusions numericall, as shown in he able. decreases wihou bound. increases wihou bound. f 00 0 0 0 00.9997.97.5 0.5.97.9997 f approaches. f approaches. NOTE or The saemen f L, means ha he i eiss and he i is equal o L. f L The able suggess ha he value of f approaches as increases wihou bound. Similarl, f approaches as decreases wihou bound. These is a infini are denoed b and f f. Limi a negaive infini Limi a posiive infini To sa ha a saemen is rue as increases wihou bound means ha for some (large) real number M, he saemen is rue for all in he inerval : > M. The following definiion uses his concep. L f() = L M f is wihin unis of L as. Figure. ε ε Definiion of Limis a Infini Le L be a real number.. The saemen f L means ha for each > 0 here eiss an M > 0 such ha f L < whenever > M.. The saemen f L means ha for each > 0 here eiss an N < 0 such ha whenever < N. f L < The definiion of a i a infini is shown in Figure.. In his figure, noe ha for a given posiive number here eiss a posiive number M such ha, for > M, he graph of f will lie beween he horizonal lines given b L and L.
0_005.qd //0 : PM Page 99 SECTION.5 Limis a Infini 99 EXPLORATION Use a graphing uili o graph f. Describe all he imporan feaures of he graph. Can ou find a single viewing window ha shows all of hese feaures clearl? Eplain our reasoning. Wha are he horizonal asmpoes of he graph? How far o he righ do ou have o move on he graph so ha he graph is wihin 0.00 uni of is horizonal asmpoe? Eplain our reasoning. Horizonal Asmpoes In Figure., he graph of f approaches he line L as increases wihou bound. The line L is called a horizonal asmpoe of he graph of f. Definiion of a Horizonal Asmpoe The line L is a horizonal asmpoe of he graph of f if f L or f L. Noe ha from his definiion, i follows ha he graph of a funcion of can have a mos wo horizonal asmpoes one o he righ and one o he lef. Limis a infini have man of he same properies of is discussed in Secion.. For eample, if f and g boh eis, hen and f g f g f g f g. Similar properies hold for is a. When evaluaing is a infini, he following heorem is helpful. (A proof of his heorem is given in Appendi A.) THEOREM.0 Limis a Infini If r is a posiive raional number and c is an real number, hen Furhermore, if r is defined when < 0, hen c r 0. c r 0. EXAMPLE Finding a Limi a Infini Find he i: Soluion 5. Using Theorem.0, ou can wrie 5 5 5 0 5. Proper of is
0_005.qd //0 : PM Page 00 00 CHAPTER Applicaions of Differeniaion EXAMPLE Finding a Limi a Infini Find he i:. Soluion Noe ha boh he numeraor and he denominaor approach infini as approaches infini. NOTE When ou encouner an indeerminae form such as he one in Eample, ou should divide he numeraor and denominaor b he highes power of in he denominaor. 5 5 f() = + is a horizonal asmpoe. Figure.5 This resuls in an indeerminae form. To resolve his problem, ou can divide, boh he numeraor and he denominaor b. Afer dividing, he i ma be evaluaed as shown. 0 0 Divide numeraor and denominaor b. Simplif. Take is of numeraor and denominaor. Appl Theorem.0. So, he line is a horizonal asmpoe o he righ. B aking he i as, ou can see ha is also a horizonal asmpoe o he lef. The graph of he funcion is shown in Figure.5. 0 0 80 As increases, he graph of f moves closer and closer o he line. Figure. TECHNOLOGY You can es he reasonableness of he i found in Eample b evaluaing f for a few large posiive values of. For insance, f 00.970, f 000.9970, and f 0,000.9997. Anoher wa o es he reasonableness of he i is o use a graphing uili. For insance, in Figure., he graph of f is shown wih he horizonal line. Noe ha as increases, he graph of f moves closer and closer o is horizonal asmpoe.
0_005.qd //0 : PM Page 0 SECTION.5 Limis a Infini 0 A Comparison of Three Raional Funcions EXAMPLE Find each i. 5 a. 5 b. 5 c. The Granger Collecion Soluion In each case, aemping o evaluae he i produces he indeerminae form. a. Divide boh he numeraor and he denominaor b. 5 5 0 0 0 0 0 b. Divide boh he numeraor and he denominaor b. 5 5 0 0 MARIA AGNESI (78 799) Agnesi was one of a handful of women o receive credi for significan conribuions o mahemaics before he wenieh cenur. In her earl wenies, she wroe he firs e ha included boh differenial and inegral calculus. B age 0, she was an honorar member of he facul a he Universi of Bologna. c. Divide boh he numeraor and he denominaor b. 5 5 You can conclude ha he i does no eis because he numeraor increases wihou bound while he denominaor approaches. Guidelines for Finding Limis a ± of Raional Funcions. If he degree of he numeraor is less han he degree of he denominaor, hen he i of he raional funcion is 0.. If he degree of he numeraor is equal o he degree of he denominaor, hen he i of he raional funcion is he raio of he leading coefficiens.. If he degree of he numeraor is greaer han he degree of he denominaor, hen he i of he raional funcion does no eis. f() = + Use hese guidelines o check he resuls in Eample. These is seem reasonable when ou consider ha for large values of, he highes-power erm of he raional funcion is he mos influenial in deermining he i. For insance, he i as approaches infini of he funcion f() = 0 f() = 0 f has a horizonal asmpoe a 0. Figure.7 FOR FURTHER INFORMATION For more informaion on he conribuions of women o mahemaics, see he aricle Wh Women Succeed in Mahemaics b Mona Fabrican, Slvia Sviak, and Paricia Clark Kenschaf in Mahemaics Teacher. To view his aricle, go o he websie www.maharicles.com. f is 0 because he denominaor overpowers he numeraor as increases or decreases wihou bound, as shown in Figure.7. The funcion shown in Figure.7 is a special case of a pe of curve sudied b he Ialian mahemaician Maria Gaeana Agnesi. The general form of his funcion is f 8a a Wich of Agnesi and, hrough a misranslaion of he Ialian word veréré, he curve has come o be known as he Wich of Agnesi. Agnesi s work wih his curve firs appeared in a comprehensive e on calculus ha was published in 78.
0_005.qd //0 : PM Page 0 0 CHAPTER Applicaions of Differeniaion In Figure.7, ou can see ha he funcion f approaches he same horizonal asmpoe o he righ and o he lef. This is alwas rue of raional funcions. Funcions ha are no raional, however, ma approach differen horizonal asmpoes o he righ and o he lef. This is demonsraed in Eample. EXAMPLE A Funcion wih Two Horizonal Asmpoes Find each i. a. b. Soluion a. For > 0, ou can wrie. So, dividing boh he numeraor and he denominaor b produces and ou can ake he i as follows. =, Horizonal asmpoe o he righ =, Horizonal asmpoe o he lef f() = + Funcions ha are no raional ma have differen righ and lef horizonal asmpoes. Figure.8 8 The horizonal asmpoe appears o be he line bu i is acuall he line. Figure.9 8 b. For < 0, ou can wrie. So, dividing boh he numeraor and he denominaor b produces and ou can ake he i as follows. 0 0 0 0 The graph of f is shown in Figure.8. TECHNOLOGY PITFALL If ou use a graphing uili o help esimae a i, be sure ha ou also confirm he esimae analicall he picures shown b a graphing uili can be misleading. For insance, Figure.9 shows one view of he graph of 000 000 000. From his view, one could be convinced ha he graph has as a horizonal asmpoe. An analical approach shows ha he horizonal asmpoe is acuall. Confirm his b enlarging he viewing window on he graphing uili.
0_005.qd //0 : PM Page 0 SECTION.5 Limis a Infini 0 In Secion. (Eample 9), ou saw how he Squeeze Theorem can be used o evaluae is involving rigonomeric funcions. This heorem is also valid for is a infini. EXAMPLE 5 Limis Involving Trigonomeric Funcions = π = f() = sin sin = 0 As increases wihou bound, f approaches 0. Figure.0 Find each i. a. sin b. Soluion a. As approaches infini, he sine funcion oscillaes beween and. So, his i does no eis. b. Because sin, i follows ha for > 0, sin where 0 and 0. So, b he Squeeze Theorem, ou can obain sin 0 as shown in Figure.0. sin EXAMPLE Ogen Level in a Pond Suppose ha f measures he level of ogen in a pond, where f is he normal (unpollued) level and he ime is measured in weeks. When 0, organic wase is dumped ino he pond, and as he wase maerial oidizes, he level of ogen in he pond is f. Wha percen of he normal level of ogen eiss in he pond afer week? Afer weeks? Afer 0 weeks? Wha is he i as approaches infini? Ogen level.00 0.75 0.50 0.5 f() (, 0.) (, 0.5) f() = 8 0 Weeks (0, 0.9) + + The level of ogen in a pond approaches he normal level of as approaches. Figure. Soluion When,, and 0, he levels of ogen are as shown. week weeks 0 weeks To find he i as approaches infini, divide he numeraor and he denominaor b o obain f f f0 0 0 0 0 0 00%. 0 See Figure.. 50% 0% 5 9 90.% 0
0_005.qd //0 : PM Page 0 0 CHAPTER Applicaions of Differeniaion Infinie Limis a Infini Man funcions do no approach a finie i as increases (or decreases) wihou bound. For insance, no polnomial funcion has a finie i a infini. The following definiion is used o describe he behavior of polnomial and oher funcions a infini. NOTE Deermining wheher a funcion has an infinie i a infini is useful in analzing he end behavior of is graph. You will see eamples of his in Secion. on curve skeching. Definiion of Infinie Limis a Infini Le f be a funcion defined on he inerval a,.. The saemen f means ha for each posiive number M, here is a corresponding number N > 0 such ha f > M whenever > N.. The saemen f means ha for each negaive number M, here is a corresponding number N > 0 such ha f < M whenever > N. Similar definiions can be given for he saemens f. f and EXAMPLE 7 Finding Infinie Limis a Infini f() = Figure. Find each i. a. b. Soluion a. As increases wihou bound, also increases wihou bound. So, ou can wrie. b. As decreases wihou bound, also decreases wihou bound. So, ou can wrie. The graph of f in Figure. illusraes hese wo resuls. These resuls agree wih he Leading Coefficien Tes for polnomial funcions as described in Secion P.. EXAMPLE 8 Finding Infinie Limis a Infini f() = + 9 Figure. 9 = Find each i. a. b. Soluion One wa o evaluae each of hese is is o use long division o rewrie he improper raional funcion as he sum of a polnomial and a raional funcion. a. b. The saemens above can be inerpreed as saing ha as approaches ±, he funcion f behaves like he funcion g. In Secion., ou will see ha his is graphicall described b saing ha he line is a slan asmpoe of he graph of f, as shown in Figure..
0_005.qd //0 : PM Page 05 SECTION.5 Limis a Infini 05 In Eercises and, describe in our own words wha he saemen means.. f. In Eercises 8, mach he funcion wih one of he graphs [(a), (b),, (d), (e), or (f)] using horizonal asmpoes as an aid. (a) (e) Eercises for Secion.5 (b) (d) (f). f. f 5. f. 7. f sin 8. Numerical and Graphical Analsis In Eercises 9, use a graphing uili o complee he able and esimae he i as approaches infini. Then use a graphing uili o graph he funcion and esimae he i graphicall. f 0 0 8 0 0 9. f 0. 0 0 f f f 5 0 5 0 f. f 5. f. f 5. In Eercises 5 and, find h, if possible. 5. f 5 0. (a) f 5 7 (a) (b) (b) In Eercises 7 0, find each i, if possible. 7. (a) 8. (a) (b) 5 5 9. (a) 0. (a) 5 (b) See www.calccha.com for worked-ou soluions o odd-numbered eercises. h f h f h f h f 5 h f h f f In Eercises, find he i... 9 7.. 5 5.. 7. 8. 9. 0. sin cos.... cos sin (b) (b) 8 5 5
0_005.qd //0 : PM Page 0 0 CHAPTER Applicaions of Differeniaion In Eercises 5 8, use a graphing uili o graph he funcion and idenif an horizonal asmpoes. 5.. f f 7. f 8. In Eercises 9 and 0, find he i. Hin: Le / and find he i as 0. 9. sin 0. In Eercises, find he i. (Hin: Trea he epression as a fracion whose denominaor is, and raionalize he numeraor.) Use a graphing uili o verif our resul..... 5.. Numerical, Graphical, and Analic Analsis In Eercises 7 50, use a graphing uili o complee he able and esimae he i as approaches infini. Then use a graphing uili o graph he funcion and esimae he i. Finall, find he i analicall and compare our resuls wih he esimaes. f 0 0 0 0 0 7. f 8. f 9. f sin 50. 0 Wriing Abou Conceps f 9 an 9 0 5 0 f 5. The graph of a funcion f is shown below. To prin an enlarged cop of he graph, go o he websie www.mahgraphs.com. f (a) Skech f. (b) Use he graphs o esimae f and f. Eplain he answers ou gave in par (b). Wriing Abou Conceps (coninued) 5. Skech a graph of a differeniable funcion f ha saisfies he following condiions and has as is onl criical number. f < 0 for < f f 5. Is i possible o skech a graph of a funcion ha saisfies he condiions of Eercise 5 and has no poins of inflecion? Eplain. 5. If f is a coninuous funcion such ha f 5, find, if possible, f for each specified condiion. (a) The graph of f is smmeric o he -ais. (b) The graph of f is smmeric o he origin. In Eercises 55 7, skech he graph of he funcion using erema, inerceps, smmer, and asmpoes. Then use a graphing uili o verif our resul. 55. 5. 57. 58. 9 59. 0. 9 9.... 5.. 7. 8. 9. 70. f > 0 for > 7. 7. In Eercises 7 8, use a compuer algebra ssem o analze he graph of he funcion. Label an erema and/or asmpoes ha eis. f 7. f 5 7. 75. f 7. f 77. f 78. 79. f 80. g 8. g sin, > 8. f f sin
0_005.qd //0 : PM Page 07 SECTION.5 Limis a Infini 07 In Eercises 8 and 8, (a) use a graphing uili o graph f and g in he same viewing window, (b) verif algebraicall ha f and g represen he same funcion, and zoom ou sufficienl far so ha he graph appears as a line. Wha equaion does his line appear o have? (Noe ha he poins a which he funcion is no coninuous are no readil seen when ou zoom ou.) 8. 8. f f g 85. Average Cos A business has a cos of C 0.5 500 for producing unis. The average cos per uni is C C. Find he i of C as approaches infini. 8. Engine Efficienc The efficienc of an inernal combusion engine is Efficienc where v v is he raio of he uncompressed gas o he compressed gas and c is a posiive consan dependen on he engine design. Find he i of he efficienc as he compression raio approaches infini. 87. Phsics Newon s Firs Law of Moion and Einsein s Special Theor of Relaivi differ concerning a paricle s behavior as is veloci approaches he speed of ligh, c. Funcions N and E represen he prediced veloci, v, wih respec o ime,, for a paricle acceleraed b a consan force. Wrie a i saemen ha describes each heor. c v 88. Temperaure The graph shows he emperaure T, in degrees Fahrenhei, of an apple pie seconds afer i is removed from an oven and placed on a cooling rack. 7 T (0, 5) % 00 v v c N (a) Find T. Wha does his i represen? 0 (b) Find Wha does his i represen? T. E g 89. Modeling Daa The able shows he world record imes for running mile, where represens he ear, wih 0 corresponding o 900, and is he ime in minues and seconds. A model for he daa is.5. 5.70 where he seconds have been changed o decimal pars of a minue. (a) Use a graphing uili o plo he daa and graph he model. (b) Does here appear o be a iing ime for running mile? Eplain. 90. Modeling Daa The average ping speeds S (words per minue) of a ping suden afer weeks of lessons are shown in he able. S A model for he daa is S 00 > 0. 5, (a) Use a graphing uili o plo he daa and graph he model. (b) Does here appear o be a iing ping speed? Eplain. 9. Modeling Daa A hea probe is aached o he hea echanger of a heaing ssem. The emperaure T (degrees Celsius) is recorded seconds afer he furnace is sared. The resuls for he firs minues are recorded in he able. T T 5 5 :0. :07. :0. :59. 0 5 0 5 0 5..9 5.5 5. 75 90 05 0 59. 58 79 85 99 :5.5 :5. :8.9 :. :. 5 0 5 0 5 0 8 5 79 90 9 9.0.0 5. 5.0 (a) Use he regression capabiliies of a graphing uili o find a model of he form T a b c for he daa. (b) Use a graphing uili o graph T. 5 8 A raional model for he daa is T. Use a 58 graphing uili o graph he model. (d) Find T 0 and T 0. (e) Find T. (f) Inerpre he resul in par (e) in he cone of he problem. Is i possible o do his pe of analsis using T? Eplain.
0_005.qd //0 : PM Page 08 08 CHAPTER Applicaions of Differeniaion 9. Modeling Daa A conainer conains 5 liers of a 5% brine soluion. The able shows he concenraions C of he miure afer adding liers of a 75% brine soluion o he conainer. 9. The graph of f is shown. C 0 0.5.5 0.5 0.95 0. 0.5 0.9 ε f.5.5 C 0.7 0.8 0.5 0.7 ε (a) Use he regression feaures of a graphing uili o find a model of he form C a b c for he daa. (b) Use a graphing uili o graph C. 5 A raional model for hese daa is C 0. graphing uili o graph C. Use a (d) Find C and C. Which model do ou hink bes represens he concenraion of he miure? Eplain. (e) Wha is he iing concenraion? 9. A line wih slope m passes hrough he poin 0,. (a) Wrie he disance d beween he line and he poin, as a funcion of m. (b) Use a graphing uili o graph he equaion in par (a). Find dm and dm. Inerpre he resuls m m geomericall. 9. A line wih slope m passes hrough he poin 0,. (a) Wrie he disance d beween he line and he poin, as a funcion of m. (b) Use a graphing uili o graph he equaion in par (a). Find dm and dm. Inerpre he resuls m m geomericall. 95. The graph of f is shown. ε (a) Find L f. (b) Deermine and in erms of. Deermine M, where M > 0, such ha f L < for > M. (d) Deermine N, where N < 0, such ha f L < for < N. f No drawn o scale (a) Find L f and K f. (b) Deermine and in erms of. Deermine M, where M > 0, such ha f L < for > M. (d) Deermine N, where N < 0, such ha f K < for < N. 97. Consider Use he definiion of is a. infini o find values of M ha correspond o (a) 0.5 and (b) 0.. 98. Consider Use he definiion of is a. infini o find values of N ha correspond o (a) 0.5 and (b) 0.. In Eercises 99 0, use he definiion of is a infini o prove he i. 99. 00. 0 0. 0. 0 0. Prove ha if p a n n... a a 0 and q b m m... b b 0 a n 0, b m 0, hen 0, p a n, q b m ±, n < m n m. n > m No drawn o scale 0 0 0. Use he definiion of infinie is a infini o prove ha. True or False? In Eercises 05 and 0, deermine wheher he saemen is rue or false. If i is false, eplain wh or give an eample ha shows i is false. 05. If f > 0 for all real numbers, hen f increases wihou bound. 0. If f < 0 for all real numbers, hen f decreases wihou bound.