Limits and an Introduction to Calculus

Transcription

1 Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem In Matematics If a function becomes arbitraril close to a unique number L as approaces c from eiter side, te it of te function as approaces c is L. In Real Life Te fundamental concept of integral calculus is te calculation of te area of a plane region bounded b te grap of a function. For instance, in surveing, a civil engineer uses integration to estimate te areas of irregular plots of real estate. (See Eercises 9 and 50, page 897.) David Frazier/PotoEdit IN CAREERS Tere are man careers tat use it concepts. Several are listed below. Market Researcer Eercise 7, page 880 Aquatic Biologist Eercise 5, page 888 Business Economist Eercises 55 and 56, page 888 Data Analst Eercises 57 and 58, pages 888 and

2 850 Capter Limits and an Introduction to Calculus. INTRODUCTION TO LIMITS Wat ou sould learn Use te definition of it to estimate its. Determine weter its of functions eist. Use properties of its and direct substitution to evaluate its. W ou sould learn it Te concept of a it is useful in applications involving maimization. For instance, in Eercise 5 on page 858, te concept of a it is used to verif te maimum volume of an open bo. Te Limit Concept Te notion of a it is a fundamental concept of calculus. In tis capter, ou will learn ow to evaluate its and ow te are used in te two basic problems of calculus: te tangent line problem and te area problem. Eample Finding a Rectangle of Maimum Area You are given inces of wire and are asked to form a rectangle wose area is as large as possible. Determine te dimensions of te rectangle tat will produce a maimum area. Let w represent te widt of te rectangle and let l represent te lengt of te rectangle. Because w l Perimeter is. it follows tat l w, as sown in Figure.. So, te area of te rectangle is A lw ww w w. Formula for area Substitute w for l. Simplif. w Dick Lurial/FPG/Gett Images l = w FIGURE. Using tis model for area, ou can eperiment wit different values of w to see ow to obtain te maimum area. After tring several values, it appears tat te maimum area occurs wen w 6, as sown in te table. Widt, w Area, A In it terminolog, ou can sa tat te it of A as w approaces 6 is 6. Tis is written as A w w 6 w 6 w 6. Now tr Eercise 5.

3 Section. Introduction to Limits 85 Definition of Limit An alternative notation for f L is c f L as c wic is read as f approaces L as approaces c. Definition of Limit If f becomes arbitraril close to a unique number L as approaces c from eiter side, te it of f as approaces c is L. Tis is written as f L. c Eample Estimating a Limit Numericall 5 5 FIGURE. (, ) f() = Use a table to estimate numericall te it: Let f. Ten construct a table tat sows values of f for two sets of -values one set tat approaces from te left and one tat approaces from te rigt. From te table, it appears tat te closer gets to, te closer f gets to. So, ou can estimate te it to be. Figure. adds furter support for tis conclusion. Now tr Eercise f ? In Figure., note tat te grap of f is continuous. For graps tat are not continuous, finding a it can be more difficult. Eample Estimating a Limit Numericall f( ) = 0 (0, ) 5 f() = + Use a table to estimate numericall te it: 0. Let f. Ten construct a table tat sows values of f for two sets of -values one set tat approaces 0 from te left and one tat approaces 0 from te rigt. f is undefined at = 0. FIGURE f ? From te table, it appears tat te it is. Te grap sown in Figure. verifies tat te it is. Now tr Eercise 9.

4 85 Capter Limits and an Introduction to Calculus In Eample, note tat f as a it wen 0 even toug te function is not defined wen 0. Tis often appens, and it is important to realize tat te eistence or noneistence of f at c as no bearing on te eistence of te it of f as approaces c. Eample Estimating a Limit Estimate te it:. Numerical Let f. Ten construct a table tat sows values of f for two sets of -values one set tat approaces from te left and one tat approaces from te rigt. Grapical Let f. Ten sketc a grap of te function, as sown in Figure.. From te grap, it appears tat as approaces from eiter side, f approaces. So, ou can estimate te it to be f ? f? f() = + 5 f( ) = (, ) From te tables, it appears tat te it is. Now tr Eercise. FIGURE. f is undefined at =. Eample 5 Using a Grap to Find a Limit FIGURE.5 f ( ) =, 0, = Find te it of f as approaces, were f is defined as f, 0, Because f for all oter tan and because te value of f is immaterial, it follows tat te it is (see Figure.5). So, ou can write f. Te fact tat f 0 as no bearing on te eistence or value of te it as approaces. For instance, if te function were defined as f,,. te it as approaces would be te same. Now tr Eercise 7.

5 Section. Introduction to Limits 85 Limits Tat Fail to Eist Net, ou will eamine some functions for wic its do not eist. Eample 6 Comparing Left and Rigt Beavior f() = FIGURE.6 f() = f() = Sow tat te it does not eist. 0 Consider te grap of te function given b see tat for positive -values and for negative -values,, > 0 < 0. From Figure.6, ou can Tis means tat no matter ow close gets to 0, tere will be bot positive and negative -values tat ield f and f. Tis implies tat te it does not eist. Now tr Eercise. f. Eample 7 Unbounded Beavior f() = FIGURE.7 Discuss te eistence of te it. 0 Let f. In Figure.7, note tat as approaces 0 from eiter te rigt or te left, f increases witout bound. Tis means tat b coosing close enoug to 0, ou can force f to be as large as ou want. For instance, f will be larger tan 00 if ou coose tat is witin of 0. Tat is, 0 < < 0 Similarl, ou can force f to be larger tan,000,000, as follows. 0 < < Because f is not approacing a unique real number L as approaces 0, ou can conclude tat te it does not eist. Now tr Eercise. f > 00. f >,000,000

6 85 Capter Limits and an Introduction to Calculus Eample 8 Oscillating Beavior f() = sin Discuss te eistence of te it. 0 sin Let f sin. In Figure.8, ou can see tat as approaces 0, f oscillates between and. Terefore, te it does not eist because no matter ow close ou are to 0, it is possible to coose values of and suc tat sin and sin, as indicated in te table sin? FIGURE.8 Now tr Eercise 5. Eamples 6, 7, and 8 sow tree of te most common tpes of beavior associated wit te noneistence of a it. Conditions Under Wic Limits Do Not Eist Te it of f as c does not eist if an of te following conditions are true.. f approaces a different number from te rigt side of c tan it approaces from te left side of c.. f increases or decreases witout bound as approaces c.. f oscillates between two fied values as approaces c. Eample 6 Eample 7 Eample 8 f() = sin FIGURE.9 TECHNOLOGY A graping utilit can elp ou discover te beavior of a function near te -value at wic ou are tring to evaluate a it. Wen ou do tis, owever, ou sould realize tat ou can t alwas trust te graps tat graping utilities displa. For instance, if ou use a graping utilit to grap te function in Eample 8 over an interval containing 0, ou will most likel obtain an incorrect grap, as sown in Figure.9. Te reason tat a graping utilit can t sow te correct grap is tat te grap as infinitel man oscillations over an interval tat contains 0.

7 Section. Introduction to Limits 855 Properties of Limits and Direct Substitution You ave seen tat sometimes te it of f as c is simpl f c, as sown in Eample. In suc cases, it is said tat te it can be evaluated b direct substitution. Tat is, f f c). c Substitute c for. Tere are man well-beaved functions, suc as polnomial functions and rational functions wit nonzero denominators, tat ave tis propert. Some of te basic ones are included in te following list. Basic Limits Let b and c be real numbers and let n be a positive integer.. c Limit of a constant function. c Limit of te identit function. c n c n Limit of a power function. n c, n c for n even and c > 0 Limit of a radical function For a proof of te it of a power function, see Proofs in Matematics on page 906. Trigonometric functions can also be included in tis list. For instance, and sin sin 0 cos cos 0. 0 B combining te basic its wit te following operations, ou can find its for a wide variet of functions. Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions wit te following its. f L c. Scalar multiple:. Sum or difference:. Product: and g K c b f bl c f ± g L ± K c fg LK c f. Quotient: provided K 0 c g L K, 5. Power: c f n L n

8 856 Capter Limits and an Introduction to Calculus Eample 9 Direct Substitution and Properties of Limits FIGURE.0 (, 6) = Find eac it. a. b. c. d. e. cos f. You can use te properties of its and direct substitution to evaluate eac it. a. 9 b. 5 5 Propert tan tan c. Propert d. 9 9 e. cos cos Propert cos 5 f. Properties and Now tr Eercise 7. Wen evaluating its, remember tat tere are several was to solve most problems. Often, a problem can be solved numericall, grapicall, or algebraicall. Te its in Eample 9 were found algebraicall. You can verif te solutions numericall and/or grapicall. For instance, to verif te it in Eample 9(a) numericall, create a table tat sows values of for two sets of -values one set tat approaces from te left and one tat approaces from te rigt, as sown below. From te table, ou can see tat te it as approaces is 6. Now, to verif te it grapicall, sketc te grap of. From te grap sown in Figure.0, ou can determine tat te it as approaces is 6. tan ?

9 Section. Introduction to Limits 857 Te results of using direct substitution to evaluate its of polnomial and rational functions are summarized as follows. Limits of Polnomial and Rational Functions. If p is a polnomial function and c is a real number, ten p p c. c. If r is a rational function given b r p q, and c is a real number suc tat q c 0, ten r r c c p c. q c For a proof of te it of a polnomial function, see Proofs in Matematics on page 906. Eample 0 Evaluating Limits b Direct Substitution Find eac it. a. 6 b. 6 Te first function is a polnomial function and te second is a rational function wit a nonzero denominator at. So, ou can evaluate te its b direct substitution. a b Now tr Eercise 5. CLASSROOM DISCUSSION Graps wit Holes Sketc te grap of eac function. Ten find te its of eac function as approaces and as approaces. Wat conclusions can ou make? a. f! "! b. g! " c.!!! Use a graping utilit to grap eac function above. Does te graping utilit distinguis among te tree graps? Write a sort eplanation of our findings.

10 858 Capter Limits and an Introduction to Calculus. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in te blanks.. If f becomes arbitraril close to a unique number L as approaces c from eiter side, te of f as approaces c is L.. An alternative notation for f L is f L as c, wic is read as f L as c. c. Te it of f as c does not eist if f between two fied values.. To evaluate te it of a polnomial function, use. SKILLS AND APPLICATIONS 5. GEOMETRY You create an open bo from a square piece of material centimeters on a side. You cut equal squares from te corners and turn up te sides. (a) Draw and label a diagram tat represents te bo. (b) Verif tat te volume V of te bo is given b V. (c) Te bo as a maimum volume wen. Use a graping utilit to complete te table and observe te beavior of te function as approaces. Use te table to find V V (d) Use a graping utilit to grap te volume function. Verif tat te volume is maimum wen. 6. GEOMETRY You are given wire and are asked to form a rigt triangle wit a potenuse of 8 inces wose area is as large as possible. (a) Draw and label a diagram tat sows te base and eigt of te triangle. (b) Verif tat te area A of te triangle is given b A 8. (c) Te triangle as a maimum area wen inces. Use a graping utilit to complete te table and observe te beavior of te function as approaces. Use te table to find A A (d) Use a graping utilit to grap te area function. Verif tat te area is maimum wen inces. In Eercises 7, complete te table and use te result to estimate te it numericall. Determine weter or not te it can be reaced f? f? f?. 0 sin f? f f? f

11 Section. Introduction to Limits 859. tan f? f.. In Eercises 6, create a table of values for te function and use te result to estimate te it numericall. Use a graping utilit to grap te corresponding function to confirm our result grapicall sin e.. 0 ln In Eercises 7 and 8, grap te function and find te it (if it eists) as approaces f,, f,, In Eercises 9 6, use te grap to find te it (if it eists). If te it does not eist, eplain w < > sin tan ln cos 0 e cos 6. 0 In Eercises 7, use a graping utilit to grap te function and use te grap to determine weter te it eists. If te it does not eist, eplain w f 5 e, f ln7, f cos, f sin, f 0 f, 5 f, f f, f 0 f. f 7 f, f f f sin

12 860 Capter Limits and an Introduction to Calculus In Eercises 5 and 6, use te given information to evaluate eac it (a) (c) (a) (c) (d) (b) (d) In Eercises 7 and 8, find (a) f, (c) [ fg], and (d) [g f]. 7. g 5 f, 8. c c f In Eercises 9 68, find te it b direct substitution e 65. sin 66. tan f, g c f c g f 5, c 5g f, 67. arcsin 68. EXPLORATION g 6 c (b) f g c g c g sin f g c f c c 6f g c 5 f 6 TRUE OR FALSE? In Eercises 69 and 70, determine weter te statement is true or false. Justif our answer. 69. Te it of a function as approaces c does not eist if te function approaces from te left of c and from te rigt of c. (b) arccos g, 70. Te it of te product of two functions is equal to te product of te its of te two functions. 7. THINK ABOUT IT From Eercises 7, select a it tat can be reaced and one tat cannot be reaced. (a) Use a graping utilit to grap te corresponding functions using a standard viewing window. Do te graps reveal weter or not te it can be reaced? Eplain. (b) Use a graping utilit to grap te corresponding functions using a decimal setting. Do te graps reveal weter or not te it can be reaced? Eplain. 7. THINK ABOUT IT Use te results of Eercise 7 to draw a conclusion as to weter or not ou can use te grap generated b a graping utilit to determine reliabl if a it can be reaced. 7. THINK ABOUT IT (a) If f, can ou conclude anting about f? Eplain our reasoning. (b) If f, can ou conclude anting about f? Eplain our reasoning. 7. WRITING Write a brief description of te meaning of te notation f. 75. THINK ABOUT IT Use a graping utilit to grap te tangent function. Wat are tan and tan? 0 Wat can ou sa about te eistence of te it tan? CAPSTONE Use te grap of te function f to decide weter te value of te given quantit eists. If it does, find it. If not, eplain w. (a) f0 (b) f 0 (c) f (d) f 77. WRITING Use a graping utilit to grap te function given b f 0. Use te trace feature 5 to approimate f. Wat do ou tink f 5 equals? Is f defined at 5? Does tis affect te eistence of te it as approaces 5? 5

13 Section. Tecniques for Evaluating Limits 86. TECHNIQUES FOR EVALUATING LIMITS Wat ou sould learn Use te dividing out tecnique to evaluate its of functions. Use te rationalizing tecnique to evaluate its of functions. Approimate its of functions grapicall and numericall. Evaluate one-sided its of functions. Evaluate its of difference quotients from calculus. W ou sould learn it Limits can be applied in real-life situations. For instance, in Eercise 8 on page 870, ou will determine its involving te costs of making potocopies. Dividing Out Tecnique In Section., ou studied several tpes of functions wose its can be evaluated b direct substitution. In tis section, ou will stud several tecniques for evaluating its of functions for wic direct substitution fails. Suppose ou were asked to find te following it. 6 Direct substitution produces 0 in bot te numerator and denominator Numerator is 0 wen. Denominator is 0 wen. Te resulting fraction, 0, as no meaning as a real number. It is called an indeterminate form because ou cannot, from te form alone, determine te it. B using a table, owever, it appears tat te it of te function as is ? Micael Krasowitz/TAXI/Gett Images Wen ou tr to evaluate a it of a rational function b direct substitution 0 and encounter te indeterminate form 0, ou can conclude tat te numerator and denominator must ave a common factor. After factoring and dividing out, ou sould tr direct substitution again. Eample sows ow ou can use te dividing out tecnique to evaluate its of tese tpes of functions. Eample Find te it: Dividing Out Tecnique 6. From te discussion above, ou know tat direct substitution fails. So, begin b factoring te numerator and dividing out an common factors. 6 5 Now tr Eercise. Factor numerator. Divide out common factor. Simplif. Direct substitution and simplif.

14 86 Capter Limits and an Introduction to Calculus Te validit of te dividing out tecnique stems from te fact tat if two functions agree at all but a single number c, te must ave identical it beavior at c. In Eample, te functions given b f 6 and g agree at all values of oter tan. So, ou can use g to find te it of f. Eample Dividing Out Tecnique FIGURE. f( ) = (, ) + f is undefined wen =. Find te it. Begin b substituting into te numerator and denominator. 0 0 Numerator is 0 wen. Denominator is 0 wen. Because bot te numerator and denominator are zero wen, direct substitution will not ield te it. To find te it, ou sould factor te numerator and denominator, divide out an common factors, and ten tr direct substitution again. Tis result is sown grapicall in Figure.. Now tr Eercise 5. Factor denominator. Divide out common factor. Simplif. Direct substitution Simplif. In Eample, te factorization of te denominator can be obtained b dividing b or b grouping as follows.

15 Section. Tecniques for Evaluating Limits 86 You can review te tecniques for rationalizing numerators and denominators in Appendi A.. Rationalizing Tecnique Anoter wa to find te its of some functions is first to rationalize te numerator of te function. Tis is called te rationalizing tecnique. Recall tat rationalizing te numerator means multipling te numerator and denominator b te conjugate of te numerator. For instance, te conjugate of is. Eample Rationalizing Tecnique Find te it: 0. ( ) FIGURE. f( ) = 0, + f is undefined wen = 0. B direct substitution, ou obtain te indeterminate form Indeterminate form In tis case, ou can rewrite te fraction b rationalizing te numerator., Now ou can evaluate te it b direct substitution. Multipl. Simplif. Divide out common factor. Simplif You can reinforce our conclusion tat te it is b constructing a table, as sown below, or b sketcing a grap, as sown in Figure.. Now tr Eercise f ? Te rationalizing tecnique for evaluating its is based on multiplication b a convenient form of. In Eample, te convenient form is.

16 86 Capter Limits and an Introduction to Calculus Using Tecnolog Te dividing out and rationalizing tecniques ma not work well for finding its of nonalgebraic functions. You often need to use more sopisticated analtic tecniques to find its of tese tpes of functions. Eample Approimating a Limit Approimate te it: 0. Numerical Let f. Because ou are finding te it wen 0, use te table feature of a graping utilit to create a table tat sows te values of f for starting at 0.0 and as a step of 0.00, as sown in Figure.. Because 0 is alfwa between 0.00 and 0.00, use te average of te values of f at tese two -coordinates to estimate te it, as follows Te actual it can be found algebraicall to be e.788. Grapical To approimate te it grapicall, grap te function f, as sown in Figure.. Using te zoom and trace features of te graping utilit, coose two points on te grap of f, suc as ,.785 and as sown in Figure.5. Because te -coordinates of tese two points are equidistant from 0, ou can approimate te it to be te average of te -coordinates. Tat is, Te actual it can be found algebraicall to be e f() = ( + )/ , FIGURE. Now tr Eercise 7. 0 FIGURE. FIGURE Eample 5 Approimating a Limit Grapicall Approimate te it: sin 0. FIGURE.6 f() = sin Direct substitution produces te indeterminate form 0. To approimate te it, begin b using a graping utilit to grap f sin, as sown in Figure.6. Ten use te zoom and trace features of te graping utilit to coose a point on eac side of 0, suc as , and , Finall, approimate te it as te average of te -coordinates of tese two points, sin It can be sown algebraicall tat tis it is eactl. 0 Now tr Eercise. 0

17 Section. Tecniques for Evaluating Limits 865 TECHNOLOGY Te graps sown in Figures. and.6 appear to be continuous at 0. However, wen ou tr to use te trace or te value feature of a graping utilit to determine te value of wen 0, no value is given. Some graping utilities can sow breaks or oles in a grap wen an appropriate viewing window is used. Because te oles in te graps in Figures. and.6 occur on te -ais, te oles are not visible. One-Sided Limits In Section., ou saw tat one wa in wic a it can fail to eist is wen a function approaces a different value from te left side of c tan it approaces from te rigt side of c. Tis tpe of beavior can be described more concisel wit te concept of a one-sided it. c f L c f L or f L as c Limit from te left or f L as c Limit from te rigt Eample 6 Evaluating One-Sided Limits f() = FIGURE.7 f() = f ( ) = Find te it as 0 from te left and te it as 0 from te rigt for f. From te grap of f, sown in Figure.7, ou can see tat f for all < 0. Terefore, te it from te left is. 0 Limit from te left: f as 0 Because f for all > 0, te it from te rigt is. 0 Now tr Eercise 55. Limit from te rigt: f as 0 In Eample 6, note tat te function approaces different its from te left and from te rigt. In suc cases, te it of f as c does not eist. For te it of a function to eist as c, it must be true tat bot one-sided its eist and are equal. Eistence of a Limit If f is a function and c and L are real numbers, ten f L c if and onl if bot te left and rigt its eist and are equal to L.

18 866 Capter Limits and an Introduction to Calculus Eample 7 Finding One-Sided Limits Find te it of f as approaces. f,, < > FIGURE.8 f() =, < f() =, > Remember tat ou are concerned about te value of f near rater tan at. So, for <, f is given b, and ou can use direct substitution to obtain For >, f is given b, and ou can use direct substitution to obtain f. f. Because te one-sided its bot eist and are equal to, it follows tat f. Te grap in Figure.8 confirms tis conclusion. Now tr Eercise 59. Eample 8 Comparing Limits from te Left and Rigt Sipping cost (in dollars) Overnigt Deliver For <, f() = 0 9 For <, f() = For 0 <, f() = 8 Weigt (in pounds) FIGURE.9 To sip a package overnigt, a deliver service carges $8 for te first pound and $ for eac additional pound or portion of a pound. Let represent te weigt of a package and let f represent te sipping cost. Sow tat te it of f as does not eist. f $8, $0, $, 0 < < < Te grap of f is sown in Figure.9. Te it of f as approaces from te left is f 0 wereas te it of f as approaces from te rigt is f. Because tese one-sided its are not equal, te it of f as does not eist. Now tr Eercise 8.

19 Section. Tecniques for Evaluating Limits 867 A Limit from Calculus In te net section, ou will stud an important tpe of it from calculus te it of a difference quotient. Eample 9 Evaluating a Limit from Calculus For te function given b f, find 0 f f. Direct substitution produces an indeterminate form. f f B factoring and dividing out, ou obtain te following. 0 f f So, te it is 6. Now tr Eercise For a review of evaluating difference quotients, refer to Section.. Note tat for an -value, te it of a difference quotient is an epression of te form 0 f f. Direct substitution into te difference quotient alwas produces te indeterminate form For instance, f f f 0 f f f

20 868 Capter Limits and an Introduction to Calculus. EXERCISES VOCABULARY: Fill in te blanks. See for worked-out solutions to odd-numbered eercises.. To evaluate te it of a rational function tat as common factors in its numerator and denominator, use te Te fraction as no meaning as a real number and terefore is called an.. Te it f L is an eample of a. c. Te it of a is an epression of te form SKILLS AND APPLICATIONS 0 f f. In Eercises 5 8, use te grap to determine eac it visuall (if it eists). Ten identif anoter function tat agrees wit te given function at all but one point g 6 6 (a) g 0 (b) g (c) g (a) g (b) g (c) g 0 (a) (b) (c) f g (c) 6 (a) f (b) f 0 f In Eercises 9 6, find te it (if it eists). Use a graping utilit to verif our result grapicall t t t sec csc.. 0 tan cot sin cos a 6 a a 7 z 7 z 0 z cos sin

21 Section. Tecniques for Evaluating Limits 869 In Eercises 7 8, use a graping utilit to grap te function and approimate te it accurate to tree decimal places. e ln 0. sin.. 0 tan GRAPHICAL, NUMERICAL, AND ALGEBRAIC ANALYSIS In Eercises 9 5, (a) grapicall approimate te it (if it eists) b using a graping utilit to grap te function, (b) numericall approimate te it (if it eists) b using te table feature of a graping utilit to create a table, and (c) algebraicall evaluate te it (if it eists) b te appropriate tecnique(s) In Eercises 55 6, grap te function. Determine te it (if it eists) b evaluating te corresponding one-sided its f were 60. f were 6. f were 6. f were 0 f,, f,, f,, f,, e 0 0 ln sin 0 cos > < > 0 > 0 In Eercises 6 68, use a graping utilit to grap te function and te equations and in te same viewing window. Use te grap to find f In Eercises 69 and 70, state wic it can be evaluated b using direct substitution. Ten evaluate or approimate eac it. 69. (a) (b) 70. (a) (b) In Eercises 7 78, find f cos f sin f sin f cos f sin f cos 0 sin sin 0 0 cos cos 0 f f f f f f f f FREE-FALLING OBJECT position function st 6t 56 0 f f. In Eercises 79 and 80, use te wic gives te eigt (in feet) of a free-falling object. Te velocit at time t a seconds is given b /a t. t a 79. Find te velocit wen t second. 80. Find te velocit wen t seconds.

22 870 Capter Limits and an Introduction to Calculus 8. SALARY CONTRACT A union contract guarantees an 8% salar increase earl for ears. For a current salar of $0,000, te salaries ft (in tousands of dollars) for te net ears are given b ft 0.000,.00,.99, were t represents te time in ears. Sow tat te it of f as t does not eist. 8. CONSUMER AWARENESS Te cost of sending a package overnigt is $5 for te first pound and $.0 for eac additional pound or portion of a pound. A plastic mailing bag can old up to pounds. Te cost f of sending a package in a plastic mailing bag is given b f 5.00, 6.0, 7.60, were represents te weigt of te package (in pounds). Sow tat te it of f as does not eist. 8. CONSUMER AWARENESS Te cost of ooking up and towing a car is $85 for te first mile and $5 for eac additional mile or portion of a mile. A model for te cost C (in dollars) is C 85 5, were is te distance in miles. (Recall from Section.6 tat f te greatest integer less tan or equal to. ) (a) Use a graping utilit to grap C for 0 < 0. (b) Complete te table and observe te beavior of C as approaces 5.5. Use te grap from part (a) and te table to find C. 5.5 (c) Complete te table and observe te beavior of C as approaces 5. Does te it of C as approaces 5 eist? Eplain. 8. CONSUMER AWARENESS Te cost C (in dollars) of making potocopies at a cop sop is given b te function C 0.5, 0.0, 0.07, 0.05, 0 < t < t < t 0 < < < C? C? 0 < 5 5 < < 500 > 500 (a) Sketc a grap of te function. (b) Find eac it and interpret our result in te contet of te situation. (i) (ii) (iii) (c) Create a table of values to sow numericall tat eac it does not eist. (i) (ii) (iii) (d) Eplain ow ou can use te grap in part (a) to verif tat te its in part (c) do not eist. EXPLORATION TRUE OR FALSE? In Eercises 85 and 86, determine weter te statement is true or false. Justif our answer. 85. Wen our attempt to find te it of a rational function 0 ields te indeterminate form 0, te rational function s numerator and denominator ave a common factor. 86. If fc L, ten f L. c 87. THINK ABOUT IT (a) Sketc te grap of a function for wic f is defined but for wic te it of f as approaces does not eist. (b) Sketc te grap of a function for wic te it of f as approaces is but for wic f. 89. WRITING Consider te it of te rational function given b pq. Wat conclusion can ou make if direct substitution produces eac epression? Write a sort paragrap eplaining our reasoning. (a) (b) (c) 5 C 5 C 88. CAPSTONE Given f,, p c q 0 p c q p c q 0 p (d) c q > 0, 99 C 00 C C 05 C 500 find eac of te following its. If te it does not eist, eplain w. (a) f (b) f (c) f 0 0 0

23 Section. Te Tangent Line Problem 87. THE TANGENT LINE PROBLEM Wat ou sould learn Use a tangent line to approimate te slope of a grap at a point. Use te it definition of slope to find eact slopes of graps. Find derivatives of functions and use derivatives to find slopes of graps. W ou sould learn it Te slope of te grap of a function can be used to analze rates of cange at particular points on te grap. For instance, in Eercise 7 on page 880, te slope of te grap is used to analze te rate of cange in book sales for particular selling prices. Tangent Line to a Grap Calculus is a branc of matematics tat studies rates of cange of functions. If ou go on to take a course in calculus, ou will learn tat rates of cange ave man applications in real life. Earlier in te tet, ou learned ow te slope of a line indicates te rate at wic a line rises or falls. For a line, tis rate (or slope) is te same at ever point on te line. For graps oter tan lines, te rate at wic te grap rises or falls canges from point to point. For instance, in Figure.0, te parabola is rising more quickl at te point, tan it is at te point,. At te verte,, te grap levels off, and at te point,, te grap is falling. (, ) (, ) (, ) (, ) FIGURE.0 Bob Rowan, Progressive Image/Corbis To determine te rate at wic a grap rises or falls at a single point, ou can find te slope of te tangent line at tat point. In simple terms, te tangent line to te grap of a function f at a point P, is te line tat best approimates te slope of te grap at te point. Figure. sows oter eamples of tangent lines. P P P FIGURE. From geometr, ou know tat a line is tangent to a circle if te line intersects te circle at onl one point. Tangent lines to noncircular graps, owever, can intersect te grap at more tan one point. For instance, in te first grap in Figure., if te tangent line were etended, it would intersect te grap at a point oter tan te point of tangenc.

24 87 Capter Limits and an Introduction to Calculus Slope of a Grap Because a tangent line approimates te slope of te grap at a point, te problem of finding te slope of a grap at a point is te same as finding te slope of te tangent line at te point. Eample Visuall Approimating te Slope of a Grap f() = 5 FIGURE. Use te grap in Figure. to approimate te slope of te grap of f at te point,. From te grap of f, ou can see tat te tangent line at, rises approimatel two units for eac unit cange in. So, ou can estimate te slope of te tangent line at, to be Slope cange in cange in. Because te tangent line at te point, as a slope of about, ou can conclude tat te grap of f as a slope of about at te point,. Now tr Eercise 5. Wen ou are visuall approimating te slope of a grap, remember tat te scales on te orizontal and vertical aes ma differ. Wen tis appens (as it frequentl does in applications), te slope of te tangent line is distorted, and ou must be careful to account for te difference in scales. Eample Approimating te Slope of a Grap Temperature ( F) FIGURE. Montl Normal Temperatures Mont (0, 69) Figure. grapicall depicts te montl normal temperatures (in degrees Fareneit) for Dallas, Teas. Approimate te slope of tis grap at te indicated point and give a psical interpretation of te result. (Source: National Catic Data Center) From te grap, ou can see tat te tangent line at te given point falls approimatel 6 units for eac two-unit cange in. So, ou can estimate te slope at te given point to be Slope cange in cange in 6 8 degrees per mont. Tis means tat ou can epect te montl normal temperature in November to be about 8 degrees lower tan te normal temperature in October. Now tr Eercise 7.

25 Section. Te Tangent Line Problem 87 ( +, f( + )) (, f( )) FIGURE. f( + ) f( ) Slope and te Limit Process In Eamples and, ou approimated te slope of a grap at a point b creating a grap and ten eeballing te tangent line at te point of tangenc. A more precise metod of approimating tangent lines makes use of a secant line troug te point of tangenc and a second point on te grap, as sown in Figure.. If, f is te point of tangenc and, f is a second point on te grap of f, te slope of te secant line troug te two points is given b m sec cange in cange in f f. Slope of secant line Te rigt side of tis equation is called te difference quotient. Te denominator is te cange in, and te numerator is te cange in. Te beaut of tis procedure is tat ou obtain more and more accurate approimations of te slope of te tangent line b coosing points closer and closer to te point of tangenc, as sown in Figure.5. ( +, f( + )) ( +, f( + )) ( +, f( + )) (, f( )) f( + ) f( ) (, f( )) f( + ) f( ) (, f( )) f( + ) f( ) Tangent line (, f( )) As approaces 0, te secant line approaces te tangent line. FIGURE.5 Using te it process, ou can find te eact slope of te tangent line at, f. Definition of te Slope of a Grap Te slope m of te grap of f at te point, f is equal to te slope of its tangent line at, f, and is given b m 0 m sec 0 f f provided tis it eists. From te definition above and from Section., ou can see tat te difference quotient is used frequentl in calculus. Using te difference quotient to find te slope of a tangent line to a grap is a major concept of calculus.

26 87 Capter Limits and an Introduction to Calculus Eample Finding te Slope of a Grap Tangent line at (, ) m = FIGURE.6 5 f() = Find te slope of te grap of f at te point,. Find an epression tat represents te slope of a secant line at,. m sec f f, 0 Net, take te it of as approaces 0. m sec Set up difference quotient. Substitute in f. Epand terms. Simplif. Factor and divide out. Simplif. m m 0 sec 0 Te grap as a slope of at te point,, as sown in Figure.6. Now tr Eercise 9. Eample Finding te Slope of a Grap FIGURE.7 f() = + m = Find te slope of f. m 0 f f Set up difference quotient. Substitute in f. Epand terms. Divide out. Simplif. You know from our stud of linear functions tat te line given b f as a slope of, as sown in Figure.7. Tis conclusion is consistent wit tat obtained b te it definition of slope, as sown above. Now tr Eercise.

27 Section. Te Tangent Line Problem 875 TECHNOLOGY Tr verifing te result in Eample 5 b using a graping utilit to grap te function and te tangent lines at, and, 5 as in te same viewing window. Some graping utilities even ave a tangent feature tat automaticall graps te tangent line to a curve at a given point. If ou ave suc a graping utilit, tr verifing Eample 5 using tis feature. Tangent line at (, ) f() = + FIGURE Tangent line at (, 5) It is important tat ou see te difference between te was te difference quotients were set up in Eamples and. In Eample, ou were finding te slope of a grap at a specific point c, f c. To find te slope in suc a case, ou can use te following form of te difference quotient. m 0 f c f c Slope at specific point In Eample, owever, ou were finding a formula for te slope at an point on te grap. In suc cases, ou sould use, rater tan c, in te difference quotient. m 0 f f Formula for slope Ecept for linear functions, tis form will alwas produce a function of, wic can ten be evaluated to find te slope at an desired point. Eample 5 Finding a Formula for te Slope of a Grap Find a formula for te slope of te grap of f. Wat are te slopes at te points, and, 5? m sec, Net, take te it of as approaces 0. m 0 m sec Set up difference quotient. Substitute in f. Epand terms. Simplif. Factor and divide out. Simplif. Formula for finding slope Using te formula m for te slope at, f, ou can find te slope at te specified points. At,, te slope is m and at, 5, te slope is m. f f 0 m sec 0 Te grap of f is sown in Figure.8. Now tr Eercise 7.

28 876 Capter Limits and an Introduction to Calculus Te Derivative of a Function In Eample 5, ou started wit te function f and used te it process to derive anoter function, m, tat represents te slope of te grap of f at te point, f. Tis derived function is called te derivative of f at. It is denoted b f, wic is read as f prime of. In Section.5, ou studied te slope of a line, wic represents te average rate of cange over an interval. Te derivative of a function is a formula wic represents te instantaneous rate of cange at a point. Definition of Derivative Te derivative of f at is given b f 0 f f provided tis it eists. Remember tat te derivative f is a formula for te slope of te tangent line to te grap of f at te point, f. Eample 6 Finding a Derivative Find te derivative of f. f 0 f f So, te derivative of f is f 6. Now tr Eercise. Note tat in addition to f, oter notations can be used to denote te derivative of f. Te most common are d d, , d f, d and D.

29 Section. Eample 7 Te Tangent Line Problem 877 Using te Derivative Find f! for f,.. Ten find te slopes of te grap of f at te points, and f! 0 Remember tat in order to rationalize te numerator of an epression, ou must multipl te numerator and denominator b te conjugate of te numerator. 0 0 (, ) f() = FIGURE At te point,, te slope is m= (, ) Because direct substitution ields te indeterminate form 00, ou sould use te rationalizing tecnique discussed in Section. to find te it. f f m= f! At te point,, te slope is 5. f!. Te grap of f is sown in Figure.9..9 Now tr Eercise. CLASSROOM DISCUSSION Using a Derivative to Find Slope In man applications, it is convenient to use a variable oter tan as te independent variable. Complete te following it process to find te derivative of f t!! /t. Ten use te result to find te slope of te grap of f t!! /t at te point,!. f t "! f! t!! 0 f t! t" 0 t!... Write a sort paragrap summarizing our findings.

30 878 Capter Limits and an Introduction to Calculus. EXERCISES VOCABULARY: Fill in te blanks.. is te stud of te rates of cange of functions. See for worked-out solutions to odd-numbered eercises.. Te to te grap of a function at a point is te line tat best approimates te slope of te grap at te point.. A is a line troug te point of tangenc and a second point on te grap.. Te slope of te tangent line to a grap at, f is given b. SKILLS AND APPLICATIONS In Eercises 5 8, use te figure to approimate te slope of te curve at te point, (, ) (, ) 9. f 0. (a) 0, (b),. f. (a) 5, (b) 0, f (a) 0, (b), f (a) 5, (b) 8, (, ) (, ) In Eercises 8, sketc a grap of te function and te tangent line at te point, f. Use te grap to approimate te slope of te tangent line.. f. f 5. f 6. f 7. f 8. f In Eercises 9 6, use te it process to find te slope of te grap of te function at te specified point. Use a graping utilit to confirm our result g,., g,, f 0,, g 5,, 5,,, 9, 0,, g, In Eercises 7, find a formula for te slope of te grap of f at te point, f. Ten use it to find te slope at te two given points. 7. f 8. f (a) 0, (a), (b), (b), 8, In Eercises 9, find te derivative of te function. 9. f 5 0. f. g 9. f 5. f. f 5. f 6. f 7. f 8. f 8 9. f f. s 9 s In Eercises 50, (a) find te slope of te grap of f at te given point, (b) use te result of part (a) to find an equation of te tangent line to te grap at te point, and (c) grap te function and te tangent line.. f,,. 5. f,, f,, 6. f,, 6 f 5

31 Section. Te Tangent Line Problem In Eercises 5 5, use a graping utilit to grap f over te interval [, ] and complete te table. Compare te value of te first derivative wit a visual approimation of te slope of te grap. 5. f 5. In Eercises 55 58, find an equation of te line tat is tangent to te grap of f and parallel to te given line f, f, Function f f,, f,, 5 f,, f f 5. f 5. f f Line 0 f 0 f In Eercises 59 6, find te derivative of f. Use te derivative to determine an points on te grap of f at wic te tangent line is orizontal. Use a graping utilit to verif our results. 59. f 60. f 6 6. f 9 6. f In Eercises 6 70, use te function and its derivative to determine an points on te grap of f at wic te tangent line is orizontal. Use a graping utilit to verif our results. 6. f, 6. f, f f 65. f cos, f sin, over te interval 0, 66. f sin, f cos, over te interval 0, 67. f e, 68. f e, f e e f e e f ln, f ln, f 7. PATH OF A BALL Te pat of a ball trown b a cild is modeled b 5 f ln ln were is te eigt of te ball (in feet) and is te orizontal distance (in feet) from te point from wic te ball was trown. Using our knowledge of te slopes of tangent lines, sow tat te eigt of te ball is increasing on te interval 0, and decreasing on te interval, 5. Eplain our reasoning. 7. PROFIT Te profit P (in undreds of dollars) tat a compan makes depends on te amount (in undreds of dollars) te compan spends on advertising. Te profit function is given b P Using our knowledge of te slopes of tangent lines, sow tat te profit is increasing on te interval 0, 0 and decreasing on te interval 0, Te table sows te revenues (in millions of dollars) for eba, Inc. from 000 troug 007. (Source: eba, Inc.) Year Revenue, (a) Use te regression feature of a graping utilit to find a quadratic model for te data. Let represent te time in ears, wit 0 corresponding to 000. (b) Use a graping utilit to grap te model found in part (a). Estimate te slope of te grap wen 5 and give an interpretation of te result. (c) Use a graping utilit to grap te tangent line to te model wen 5. Compare te slope given b te graping utilit wit te estimate in part (b).

32 880 Capter Limits and an Introduction to Calculus 7. MARKET RESEARCH Te data in te table sow te number N (in tousands) of books sold wen te price per book is p (in dollars). (a) Use te regression feature of a graping utilit to find a quadratic model for te data. (b) Use a graping utilit to grap te model found in part (a). Estimate te slopes of te grap wen p $5 and p $0. (c) Use a graping utilit to grap te tangent lines to te model wen p $5 and p $0. Compare te slopes given b te graping utilit wit our estimates in part (b). (d) Te slopes of te tangent lines at p $5 and p $0 are not te same. Eplain wat tis means to te compan selling te books. EXPLORATION TRUE OR FALSE? In Eercises 75 and 76, determine weter te statement is true or false. Justif our answer. 75. Te slope of te grap of is different at ever point on te grap of f. 76. A tangent line to a grap can intersect te grap onl at te point of tangenc. In Eercises 77 80, matc te function wit te grap of its derivative. It is not necessar to find te derivative of te function. [Te graps are labeled (a), (b), (c), and (d).] (a) Price, p $0 $5 $0 $5 $0 $5 Number of books, N (b) (c) 5 (d) 77. f 78. f f f 8. THINK ABOUT IT Sketc te grap of a function wose derivative is alwas positive. 8. THINK ABOUT IT Sketc te grap of a function wose derivative is alwas negative. 8. THINK ABOUT IT Sketc te grap of a function for wic f < 0 for <, f 0 for >, and f CONJECTURE Consider te functions f and g. (a) Sketc te graps of f and on te same set of coordinate aes. (b) Sketc te graps of g and on te same set of coordinate aes. (c) Identif an pattern between te functions f and g and teir respective derivatives. Use te pattern to make a conjecture about if n, were n is an integer and n. 85. Consider te function f. (a) Use a graping utilit to grap te function. (b) Use te trace feature to approimate te coordinates of te verte of tis parabola. (c) Use te derivative of f to find te slope of te tangent line at te verte. (d) Make a conjecture about te slope of te tangent line at te verte of an arbitrar parabola. PROJECT: ADVERTISING To work an etended application analzing te amount spent on advertising in te United States, visit tis tet s website at academic.cengage.com. (Data Source: Universal McCann) f g 86. CAPSTONE Eplain ow te slope of te secant line is used to derive te slope of te tangent line and te definition of te derivative of a function f at a point, f. Include diagrams or sketces as necessar.

33 Section. Limits at Infinit and Limits of Sequences 88. LIMITS AT INFINITY AND LIMITS OF SEQUENCES Wat ou sould learn Evaluate its of functions at infinit. Find its of sequences. W ou sould learn it Finding its at infinit is useful in man tpes of real-life applications. For instance, in Eercise 58 on page 889, ou are asked to find a it at infinit to determine te number of militar reserve personnel in te future. Limits at Infinit and Horizontal Asmptotes As pointed out at te beginning of tis capter, tere are two basic problems in calculus: finding tangent lines and finding te area of a region. In Section., ou saw ow its can be used to solve te tangent line problem. In tis section and te net, ou will see ow a different tpe of it, a it at infinit, can be used to solve te area problem. To get an idea of wat is meant b a it at infinit, consider te function given b f. Te grap of f is sown in Figure.0. From earlier work, ou know tat is a orizontal asmptote of te grap of tis function. Using it notation, tis can be written as follows. f Horizontal asmptote to te left Karen Kasmauski/Corbis f Horizontal asmptote to te rigt Tese its mean tat te value of f gets arbitraril close to as decreases or increases witout bound. = f() = + Te function f is a rational function. You can review rational functions in Section.6. FIGURE.0 Definition of Limits at Infinit If f is a function and and are real numbers, te statements and f L L L Limit as approaces Limit as approaces f L denote te its at infinit. Te first statement is read te it of f as approaces is L, and te second is read te it of f as approaces is L.

34 88 Capter Limits and an Introduction to Calculus To elp evaluate its at infinit, ou can use te following definition. Limits at Infinit If r is a positive real number, ten Limit toward te rigt Furtermore, if r is defined wen < 0, ten 0. r 0. r Limit toward te left Limits at infinit sare man of te properties of its listed in Section.. Some of tese properties are demonstrated in te net eample. Eample Evaluating a Limit at Infinit Find te it. Algebraic Use te properties of its listed in Section.. 0 So, te it of f as approaces is. Now tr Eercise 9. Grapical Use a graping utilit to grap. Ten use te trace feature to determine tat as gets larger and larger, gets closer and closer to, as sown in Figure.. Note tat te line is a orizontal asmptote to te rigt. 5 = = 0 0 FIGURE. In Figure., it appears tat te line is also a orizontal asmptote to te left. You can verif tis b sowing tat. Te grap of a rational function need not ave a orizontal asmptote. If it does, owever, its left and rigt orizontal asmptotes must be te same. Wen evaluating its at infinit for more complicated rational functions, divide te numerator and denominator b te igest-powered term in te denominator. Tis enables ou to evaluate eac it using te its at infinit at te top of tis page.

35 Section. Limits at Infinit and Limits of Sequences 88 Eample Comparing Limits at Infinit Find te it as approaces for eac function. a. b. f f c. f In eac case, begin b dividing bot te numerator and denominator b igest-powered term in te denominator., te a b. 0 0 c. In tis case, ou can conclude tat te it does not eist because te numerator decreases witout bound as te denominator approaces. Now tr Eercise 9. In Eample, observe tat wen te degree of te numerator is less tan te degree of te denominator, as in part (a), te it is 0. Wen te degrees of te numerator and denominator are equal, as in part (b), te it is te ratio of te coefficients of te igest-powered terms. Wen te degree of te numerator is greater tan te degree of te denominator, as in part (c), te it does not eist. Tis result seems reasonable wen ou realize tat for large values of, te igest-powered term of a polnomial is te most influential term. Tat is, a polnomial tends to beave as its igest-powered term beaves as approaces positive or negative infinit.

36 88 Capter Limits and an Introduction to Calculus Limits at Infinit for Rational Functions Consider te rational function f ND, were N a and D b m m... n n... a 0 b 0. Te it of f as approaces positive or negative infinit is as follows. ± 0, n < m f a n, n m b m If n > m, te it does not eist. Eample Finding te Average Cost You are manufacturing greeting cards tat cost $0.50 per card to produce. Your initial investment is $5000, wic implies tat te total cost C of producing cards is given b C Te average cost C per card is given b C C Find te average cost per card wen (a) 000, (b) 0,000, and (c) 00,000. (d) Wat is te it of C as approaces infinit? Average cost per card (in dollars) 6 5 = 0.5 C C C = = Average Cost ,000 60,000 00,000 Number of cards As, te average cost per card approaces $0.50. FIGURE. a. Wen 000, te average cost per card is C b. Wen 0,000, te average cost per card is C c. Wen 00,000, te average cost per card is C $ , ,000 $ , ,000 $0.55. d. As approaces infinit, te it of C is $0.50. Te grap of C is sown in Figure.. Now tr Eercise ,000 00,000

37 Section. Limits at Infinit and Limits of Sequences 885 You can review sequences in Sections TECHNOLOGY Tere are a number of was to use a graping utilit to generate te terms of a sequence. For instance, ou can displa te first 0 terms of te sequence a n n using te sequence feature or te table feature. Limits of Sequences Limits of sequences ave man of te same properties as its of functions. For instance, consider te sequence wose nt term is a n n.,, 8, 6,,... As n increases witout bound, te terms of tis sequence get closer and closer to 0, and te sequence is said to converge to 0. Using it notation, ou can write n 0. n Te following relationsip sows ow its of functions of can be used to evaluate te it of a sequence. Limit of a Sequence Let f be a function of a real variable suc tat f L. If a n is a sequence suc tat f n a n for ever positive integer n, ten n a n L. A sequence tat does not converge is said to diverge. For instance, te terms of te sequence,,,,,... oscillate between and. Terefore, te sequence diverges because it does not approac a unique number. Eample Finding te Limit of a Sequence You can use te definition of its at infinit for rational functions on page 88 to verif te its of te sequences in Eample. Find te it of eac sequence. (Assume n begins wit.) a. b. c. a. b. c. a n b n n n n n c n n n n n n n n n 0 n n n Now tr Eercise 9. 5, 5 6, 7 7, 9 8, 9, 0,... 5, 5 8, 7, 9 0, 9, 0,... 0, 9 6, 9 6, 6, 5 00, 7,...

38 886 Capter Limits and an Introduction to Calculus In te net section, ou will encounter its of sequences suc as tat sown in Eample 5. A strateg for evaluating suc its is to begin b writing te nt term in standard rational function form. Ten ou can determine te it b comparing te degrees of te numerator and denominator, as sown on page 88. Eample 5 Finding te Limit of a Sequence Find te it of te sequence wose nt term is an 8 n n n. n 6 Algebraic Numerical Begin b writing te nt term in standard rational function form as te ratio of two polnomials. Construct a table tat sows te value of an as n becomes larger and larger, as sown below. an 8 n n n n 6 8 n n n 6n 8n n n n Write original nt term. n an , Multipl fractions. Write in standard rational form. From tis form, ou can see tat te degree of te numerator is equal to te degree of te denominator. So, te it of te sequence is te ratio of te coefficients of te igest-powered terms. From te table, ou can estimate tat as n approaces 8, an gets closer and closer to.667 ". 8n n n 8 n n Now tr Eercise 9. CLASSROOM DISCUSSION Comparing Rates of Convergence In te table in Eample 5 above, te value of 8 an approaces its it of rater slowl. (Te first term to be accurate to tree decimal places is a ) Eac of te following sequences converges to 0. Wic converges te quickest? Wic converges te slowest? W? Write a sort paragrap discussing our conclusions. a. an n b. bn n d. dn n! e. n n n! c. cn n

39 Section. Limits at Infinit and Limits of Sequences 887. EXERCISES VOCABULARY: Fill in te blanks.. A at can be used to solve te area problem in calculus. See for worked-out solutions to odd-numbered eercises.. Wen evaluating its at infinit for complicated rational functions, ou can divide te numerator and denominator b te term in te denominator.. A sequence tat as a it is said to.. A sequence tat does not ave a it is said to. SKILLS AND APPLICATIONS In Eercises 5 8, matc te function wit its grap, using orizontal asmptotes as aids. [Te graps are labeled (a), (b), (c), and (d).] (a) (c) (b) (d) 5. f 6. f 7. f 8. f In Eercises 9 8, find te it (if it eists). If te it does not eist, eplain w. Use a graping utilit to verif our result grapicall t t t 9. t 0. t t In Eercises 9, use a graping utilit to grap te function and verif tat te orizontal asmptote corresponds to te it at infinit NUMERICAL AND GRAPHICAL ANALYSIS In Eercises 5 8, (a) complete te table and numericall estimate te it as approaces infinit, and (b) use a graping utilit to grap te function and estimate te it grapicall t t t 5t t t f f 9 7. f 8. f f

40 888 Capter Limits and an Introduction to Calculus In Eercises 9 8, write te first five terms of te sequence and find te it of te sequence (if it eists). If te it does not eist, eplain w. Assume n begins wit. 9. a 0. a n n n n n n. a n n. a n n. a n. n 7. a n n 8. n In Eercises 9 5, find te it of te sequence. Ten verif te it numericall b using a graping utilit to complete te table a n n a n n n n nn a n n n n nn a n 6 n nnn 6 nn n 5. OXYGEN LEVEL Suppose tat f t measures te level of ogen in a pond, were f t is te normal (unpolluted) level and te time t is measured in weeks. Wen t 0, organic waste is dumped into te pond, and as te waste material oidizes, te level of ogen in te pond is given b f t t t t. nnn n n a n n n n! n! 5. a n 6. a n n! n! a n n n n a n (a) Wat is te it of f as t approaces infinit? (b) Use a graping utilit to grap te function and verif te result of part (a). (c) Eplain te meaning of te it in te contet of te problem. 5. TYPING SPEED Te average tping speed S (in words per minute) for a student after t weeks of lessons is given b (a) Wat is te it of S as t approaces infinit? (b) Use a graping utilit to grap te function and verif te result of part (a). (c) Eplain te meaning of te it in te contet of te problem. 55. AVERAGE COST Te cost function for a certain model of personal digital assistant (PDA) is given b C.50 5,750, were C is measured in dollars and is te number of PDAs produced. (a) Write a model for te average cost per unit produced. (b) Find te average costs per unit wen 00 and 000. (c) Determine te it of te average cost function as approaces infinit. Eplain te meaning of te it in te contet of te problem. 56. AVERAGE COST Te cost function for a compan to reccle tons of material is given b C.5 0,500, were C is measured in dollars. (a) Write a model for te average cost per ton of material reccled. (b) Find te average costs of reccling 00 tons of material and 000 tons of material. (c) Determine te it of te average cost function as approaces infinit. Eplain te meaning of te it in te contet of te problem. 57. DATA ANALYSIS: SOCIAL SECURITY Te table sows te average montl Social Securit benefits B (in dollars) for retired workers aged 6 or over from 00 troug 007. (Source: U.S. Social Securit Administration) A model for te data is given b B Year Benefit, B t.0 0.8t 0.00t, t 7 were t represents te ear, wit t corresponding to 00. S 00t 65 t, t > 0.

41 Section. Limits at Infinit and Limits of Sequences 889 (a) Use a graping utilit to create a scatter plot of te data and grap te model in te same viewing window. How well does te model fit te data? (b) Use te model to predict te average montl benefit in 0. (c) Discuss w tis model sould not be used for long-term predictions of average montl Social Securit benefits. 58. DATA ANALYSIS: MILITARY Te table sows te numbers N (in tousands) of U.S. militar reserve personnel for te ears 00 troug 007. (Source: U.S. Department of Defense) A model for te data is given b N were t represents te ear, wit t corresponding to 00. (a) Use a graping utilit to create a scatter plot of te data and grap te model in te same viewing window. How well does te model fit te data? (b) Use te model to predict te number of militar reserve personnel in 0. (c) Wat is te it of te function as t approaces infinit? Eplain te meaning of te it in te contet of te problem. Do ou tink te it is realistic? Eplain. EXPLORATION Year Number, N t, t t TRUE OR FALSE? In Eercises 59 6, determine weter te statement is true or false. Justif our answer. 59. Ever rational function as a orizontal asmptote. 60. If f increases witout bound as approaces c, ten te it of f eists. 6. If a sequence converges, ten it as a it. 6. Wen te degrees of te numerator and denominator of a rational function are equal, te it does not eist. 6. THINK ABOUT IT Find te functions f and g suc tat bot f and g increase witout bound as approaces c, but f g eists. 6. THINK ABOUT IT Use a graping utilit to grap te function given b f How man orizontal asmptotes does te function appear to ave? Wat are te orizontal asmptotes? In Eercises 65 68, create a scatter plot of te terms of te sequence. Determine weter te sequence converges or diverges. If it converges, estimate its it. a n n a n.5n Use a graping utilit to grap te two functions given b and in te same viewing window. W does not appear to te left of te -ais? How does tis relate to te statement at te top of page 88 about te infinite it r?. 7. Use a graping utilit to complete te table below to verif tat 0. c Make a conjecture about 0. a n n a n 0.5n CAPSTONE Use te grap to estimate (a) f, (b) f, and (c) te orizontal asmptote of te grap of f. (i) f 6 (ii) 6 f

42 890 Capter Limits and an Introduction to Calculus.5 THE AREA PROBLEM Adam Woolfitt/Corbis Wat ou sould learn Find its of summations. Use rectangles to approimate areas of plane regions. Use its of summations to find areas of plane regions. W ou sould learn it Te its of summations are useful in determining areas of plane regions. For instance, in Eercise 50 on page 897, ou are asked to find te it of a summation to determine te area of a parcel of land bounded b a stream and two roads. Recall from Section 9. tat te sum of a finite geometric sequence is given b n i a r i a rn Furtermore, if ten r n 0 as n. r. 0 < r <, Limits of Summations Earlier in te tet, ou used te concept of a it to obtain a formula for te sum S of an infinite geometric series S a a r a r... a r i a r, Using it notation, tis sum can be written as S n n a r i i for n rn 0 Te following summation formulas and properties are used to evaluate finite and infinite summations. Eample Evaluate te summation. Evaluating a Summation 00 i i Using te second summation formula wit n 00, ou can write n nn i i i i 0,00 0,00. a r n n r Now tr Eercise 5. i Summation Formulas and Properties n i. c cn, c is a constant... n nn (n i. 6 i n i a r. n i r < ka i k n a i ± b i n n a i ± a i, k is a constant. i b i i n i i nn n i n n i i r <