12.2 Techniques for Evaluating Limits

Transcription

1 335_qd /4/5 :5 PM Page 863 Section Tecniques for Evaluating Limits 863 Tecniques 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 7 on page 87, 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 Direct substitution produces in bot te numerator and denominator Numerator is wen Denominator is wen 3, Te resulting fraction, 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 3 is ? Wen ou tr to evaluate a it of a rational function b direct substitution and encounter te indeterminate form, 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 Dividing Out Tecnique Find te it: From te discussion above, ou know tat direct substitution fails So, begin b factoring te numerator and dividing out an common factors Micael Krasowitz/Gett Images Factor numerator Divide out common factor Simplif 3 5 Direct substitution and simplif 3 Now tr Eercise 7

2 335_qd /3/5 : PM Page 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 3 agree at all values of oter tan 3 So, ou can use g to find te it of f Consider suggesting to our students tat te tr making a table of values to estimate te it in Eample before finding it algebraicall A range of 9 troug wit increment is useful Eample Find te it Dividing Out Tecnique 3 Begin b substituting into te numerator and denominator Numerator is wen 3 Denominator is 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 3 Factor denominator Divide out common factor f () = 3 + Simplif (, ) f is undefined wen = Direct substitution Simplif FIGURE Tis result is sown grapicall in Figure Now tr Eercise 9 In Eample, te factorization of te denominator can be obtained b dividing b or b grouping as follows 3

3 335_qd /3/5 : PM Page 865 Rationalizing Tecnique Section Tecniques for Evaluating Limits 865 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 4 is 4 Eample 3 Rationalizing Tecnique 3 ( ) FIGURE f () =, + f is undefined wen = Find te it: 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 f Now tr Eercise 7 Te rationalizing tecnique for evaluating its is based on multiplication b a convenient form of In Eample 3, te convenient form is ?

4 335_qd /3/5 : PM Page 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 4 Approimating a Limit Approimate te it: Numerical Let f Because ou are finding te it wen, use te table feature of a graping utilit to create a table tat sows te value of f for starting at and as a step of, as sown in Figure 3 Because is alfwa between and, 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 4 Using te zoom and trace features of te graping utilit, coose two points on te grap of f, suc as 7, 785 and 7, 78 as sown in Figure 5 Because te -coordinates of tese two points are equidistant from, 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() = ( + )/ 75 FIGURE 3 Now tr Eercise 7 FIGURE 4 FIGURE Eample 5 Approimating a Limit Grapicall 4 4 FIGURE 6 f() = sin Approimate te it: sin Direct substitution produces te indeterminate form 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, suc as 467, and 467, 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 Now tr Eercise 3

5 335_qd /3/5 : PM Page 867 Section Tecniques for Evaluating Limits 867 Tecnolog Te graps sown in Figures 4 and 6 appear to be continuous at However, wen ou tr to use te trace or te value feature of a graping utilit to determine te value of wen, tere is no value 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 4 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 f L or f L as c Limit from te left c c f L or f L as c Limit from te rigt You migt wis to illustrate te concept of one-sided its (and w te are necessar) wit tables or graps f() = FIGURE 7 f() = f () = Eample 6 Evaluating One-Sided Limits Find te it as from te left and te it as from te rigt for f From te grap of f, sown in Figure 7, ou can see tat f for all < Terefore, te it from te left is Limit from te left: f as Because f for all >, te it from te rigt is Now tr Eercise 43 Limit from te rigt: f as 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

6 335_qd /3/5 : PM Page Capter Limits and an Introduction to Calculus Eample 7 Finding One-Sided Limits FIGURE 8 f() = 4, < f() = 4, > Find te it of f as approaces f 4, 4, Remember tat ou are concerned about te value of f near rater tan at So, for <, f is given b 4, and ou can use direct substitution to obtain For >, f is given b 4, and ou can use direct substitution to obtain f f < > Because te one-sided its bot eist and are equal to 3, it follows tat f 3 Te grap in Figure 8 confirms tis conclusion Now tr Eercise 47 Eample 8 Comparing Limits from te Left and Rigt Sipping cost (in dollars) 3 9 Overnigt Deliver For < 3, f() = For <, f() = 8 For <, f() = 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,,, < < < 3 Te grap of f is sown in Figure 9 Te it of f as approaces from te left is f 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 69

7 335_qd /3/5 : PM Page 869 A Limit from Calculus Section Tecniques for Evaluating Limits 869 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 Eample 9 previews te derivative tat is introduced in Section 3 Group Activit Write a it problem (be sure te it eists) and ecange it wit tat of a partner Use a numerical approac to estimate te it, and use an algebraic approac to verif our estimate Discuss our results wit our partner For te function given b f, find Direct substitution produces an indeterminate form f 3 f 3 f 3 f B factoring and dividing out, ou obtain te following f 3 f 3 So, te it is 6 Now tr Eercise For a review of evaluating difference quotients, refer to Section 4 Note tat for an -value, te it of a difference quotient is an epression of te form f f Direct substitution into te difference quotient alwas produces te indeterminate form For instance, f f f f f f

8 335_qd /3/5 : PM Page Capter Limits and an Introduction to Calculus Eercises VOCABULARY CHECK: Fill in te blanks 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 3 Te it f L is an eample of a c f f 4 Te it of a is an epression of te form PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for tis section at wwweduspacecom In Eercises 4, 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 (c) g 3 g 3 4 (a) g (b) (c) 6 4 (a) g (b) g g g (a) (b) (c) 3 f (a) f (b) f (c) f In Eercises 5 6, find te it (if it eists) Use a graping utilit to verif our result grapicall t 8 t t t 7 t 3 t sec 5 tan z 7 z z sin cos

9 335_qd /3/5 : PM Page 87 Section Tecniques for Evaluating Limits 87 In Eercises 7 38, use a graping utilit to grap te function and approimate te it accurate to tree decimal places 53 f sin f sin 56 f cos f cos 7 e 8 9 ln 3 3 sin 3 33 tan Grapical, Numerical, and Algebraic Analsis In Eercises 39 4, (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 43 5, grap te function Determine te it (if it eists) b evaluating te corresponding one-sided its f were 48 f were 49 f were 5 f were f, 3, f, 4, f 4, 3, f 4, 4, e ln sin 3 cos > < > > 3 In Eercises 5 56, use a graping utilit to grap te function and te equations and in te same viewing window Use te grap to find f 5 f cos 5 f sin In Eercises 57 and 58, state wic it can be evaluated b using direct substitution Ten evaluate or approimate eac it 57 (a) sin (b) 58 (a) cos (b) In Eercises 59 66, find 59 f 3 6 f f 6 f 63 f 3 64 f 4 65 f 66 f Free-Falling Object In Eercises 67 and 68, use te position function st 6t 8, wic gives te eigt (in feet) of a free-falling object Te velocit at time t a seconds is given b [sa st] /a t t a 67 Find te velocit wen t second 68 Find te velocit wen t seconds 69 Salar Contract A union contract guarantees a % salar increase earl for 3 ears For a current salar of $8,, te salar ft (in tousands of dollars) for te net 3 ears is given b ft 8, 38, 3388, were t represents te time in ears Sow tat te it of f as t does not eist 7 Consumer Awareness Te cost of sending a package overnigt is $75 for te first pound and $395 for eac additional pound or portion of a pound A plastic mailing bag can old up to 3 pounds Te cost f of sending a package in a plastic mailing bag is given b f 75, 47, 865, < t < t < t 3 < < < 3 sin cos f f were represents te weigt of te package (in pounds) Sow tat te it of f as does not eist

10 335_qd /3/5 : PM Page Capter Limits and an Introduction to Calculus 7 Communications Te cost of a telepone call between two cities is $75 for te first minute and $5 for eac additional minute or portion of a minute A model for te cost C is given b Ct 75 5t were t is te time in minutes (Recall from Section 6 tat f te greatest integer less tan or equal to ) (a) Sketc te grap of C for < t 5 (b) Complete te table and observe te beavior of C as t approaces 35 Use te grap from part (a) and te table to find (c) Complete te table and observe te beavior of C as t approaces 3 Does te it of Ct as t approaces 3 eist? Eplain Sntesis t 35 Ct t C? 7 Consumer Awareness Te cost C (in dollars) of making potocopies at a cop sop is given b te function C 5,, 7, 5, t C? (a) Sketc a grap of te function (b) Find eac it and interpret our result in te contet of te situation (i) (ii) (iii) 5 C (c) Create a table of values to sow numericall tat eac it does not eist (i) (ii) (iii) 5 C Model It < 5 5 < < 5 > 5 99 C C 35 C 5 C (d) Eplain ow ou can use te grap in part (a) to verif tat te its in part (c) do not eist True or False? In Eercises 73 and 74, determine weter te statement is true or false Justif our answer 73 Wen our attempt to find te it of a rational function ields te indeterminate form, te rational function s numerator and denominator ave a common factor 74 If fc L, ten f L 75 Tink 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 4 but for wic f 4 76 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 p (a) c q p (c) c q Skills Review c 77 Write an equation of te line tat passes troug 6, and is perpendicular to te line tat passes troug 4, 6 and 3, 4 78 Write an equation of te line tat passes troug, and is parallel to te line tat passes troug 3, 3 and 5, 79 Arc Lengt Find te lengt of te arc on a circle wit a radius of 85 inces intercepted b a central angle of 45 8 Arc Lengt Find te lengt of te arc on a circle wit a radius of 6 mileters intercepted b a central angle of 8 Circular Sector Find te area of te sector of a circle wit a radius of 9 centimeters and central angle 8 Circular Sector Find te area of te sector of a circle wit a radius of 34 feet and central angle In Eercises 83 88, identif te tpe of conic algebraicall Ten use a graping utilit to grap te conic (b) (d) 83 3 r cos 84 r 85 9 r 3 cos 86 r 87 5 r sin 88 r In Eercises 89 9, determine weter te vectors are ortogonal, parallel, or neiter 89 7,, 3,, 4, 5 9 5, 5,,, 5, 9 4, 3, 6,, 9, 8 9, 3,,,, p c q p c q 8 3 sin 4 4 cos sin 35