Limits and Continuity

Transcription

1 CHAPTER Limits and Continuit. Rates of Cange and Limits. Limits Involving Infinit.3 Continuit.4 Rates of Cange and Tangent Lines An Economic Injur Level (EIL) is a measurement of te fewest number of insect pests tat will cause economic damage to a crop or forest. It as been estimated tat monitoring pest populations and establising EILs can reduce pesticide use b 30% 50%. Accurate population estimates are crucial for determining EILs. A population densit of one insect pest can be approimated b D(t ) = t 90 + t 3 pests per plant, were t is te number of das since initial infestation. Wat is te rate of cange of tis population densit wen te population densit is equal to te EIL of 0 pests per plant? Section.4 can elp answer tis question. 58

2 Section. Rates of Cange and Limits 59 CHAPTER Overview Te concept of it is one of te ideas tat distinguis calculus from algebra and trigonometr. In tis capter, we sow ow to define and calculate its of function values. Te calculation rules are straigtforward, and most of te its we need can be found b substitution, grapical investigation, numerical approimation, algebra, or some combination of tese. One of te uses of its is to test functions for continuit. Continuous functions arise frequentl in scientific work because te model suc an enormous range of natural beavior. Te also ave special matematical properties, not oterwise guaranteed.. Rates of Cange and Limits Wat ou will learn about... Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwic Teorem and w... Limits can be used to describe continuit, te derivative, and te integral: te ideas giving te foundation of calculus. Average and Instantaneous Speed Average speed of a moving bod during an interval of time is found b dividing te distance covered b te elapsed time. More precisel, if = f(t) is a distance or position function of a moving bod at time t, ten te average rate of cange (or average speed) is te ratio were te elapsed time is te interval of time from t to t +, or simpl. Often we call te numerator, te cange in = f(t), and te denominator t, te cange in elapsed time t, were is read delta and t is read delta t. Tus we often write te ratio as t f(t + ) - f(t) = f(t + ) - f(t). Free Fall Near te surface of te eart, all bodies fall wit te same constant acceleration. Te distance a bod falls after it is released from rest is a constant multiple of te square of te time fallen. At least, tat is wat appens wen a bod falls in a vacuum, were tere is no air to slow it down. Te square-of-time rule also olds for dense, eav objects like rocks, ball bearings, and steel tools during te first few seconds of fall troug air, before te velocit builds up to were air resistance begins to matter. Wen air resistance is absent or insignificant and te onl force acting on a falling bod is te force of gravit, we call te wa te bod falls free fall. EXAMPLE Finding an Average Speed A rock breaks loose from te top of a tall cliff. Wat is its average speed during te first seconds of fall? SOLUTION Eperiments sow tat a dense solid object dropped from rest to fall freel near te surface of te eart will fall = 6t feet in te first t seconds. Te average speed of te rock over an given time interval is te distance traveled,, divided b te lengt of te interval t. For te first seconds of fall, from t = 0 to t =, we ave t = 6() - 6(0) - 0 = 3 ft sec. Now Tr Eercise. A moving bod s instantaneous speed is te speed of te moving object at a given instant of time. Te major issue is ow to compute tis speed, since te elapsed time seems to be zero, as we eplain in Eample. We sow after tis eample tat we need te matematical concept of a it to understand and compute instantaneous speed.

3 60 Capter Limits and Continuit TABLE. Average Speeds over Sort Time Intervals Starting at t t 6( + ) - 6() = Lengt of Average Speed Time Interval, for Interval (sec) > t (ft/sec) Formal Definition of Limit You ma want to look over te eamples in Appendi A3, pp , wic provide illustrations of te formal it definition of a function f tat as it L as approaces c. X [ p, p] b [, ] Y ERROR Y = sin(x)/x Figure. A grap and table of values for f() = (sin )> tat suggest te it of f as approaces 0 is. EXAMPLE Finding an Instantaneous Speed Find te speed of te rock in Eample at te instant t =. SOLUTION Solve Numericall We can calculate te average speed of te rock over te interval from time t = to an sligtl later time t = + as We cannot use tis formula to calculate te speed at te eact instant t = because tat would require taking = 0, and 0>0 is undefined. However, we can get a good idea of wat is appening at t = b evaluating te formula at values of close to 0. Wen we do, we see a clear pattern (Table.). As approaces 0, te average speed approaces te iting value 64 ft/sec. Confirm Algebraicall If we epand te numerator of Equation and simplif, we find tat t = 6( + ) - 6() = t = 6( + ) - 6(). = = 6( ) - 64 For values of different from 0, te epressions on te rigt and left are equivalent and te average speed is ft/sec. We can now see w te average speed as te iting value (0) = 64 ft/sec as approaces 0. Now Tr Eercise 3. Definition of Limit As in te preceding eample, most its of interest in te real world can be viewed as numerical its of values of functions. And tis is were a graping utilit and calculus come in. A calculator can suggest te its, and calculus can give te matematics for confirming te its analticall. Limits give us a language for describing ow te outputs of a function beave as te inputs approac some particular value. In Eample, te average speed was not defined at = 0 but approaced te it 64 as approaced 0. We were able to see tis numericall and to confirm it algebraicall b einating from te denominator. But we cannot alwas do tat. For instance, we can see bot grapicall and numericall (Figure.) tat te values of f() = (sin )> approac as approaces 0. We cannot einate te from te denominator of (sin )> to confirm te observation algebraicall. We need to use a teorem about its to make tat confirmation, as ou will see in Eercise 77. DEFINITION Limit Assume f is defined in a neigborood of c and let c and L be real numbers. Te function f as it L as approaces c if, given an positive number e, tere is a positive number d suc tat for all, We write 0 6 ƒ - c ƒ 6 d Q ƒ f() - L ƒ 6 e. f() = L. ()

4 Section. Rates of Cange and Limits 6 Te sentence f() = L is read, Te it of f of as approaces c equals L. Te notation means tat te values f() of te function f approac or equal L as te values of approac (but do not equal) c. Appendi A3 provides practice appling te definition of it. We saw in Eample tat :0 (64 + 6) = 64. As suggested in Figure., sin =. :0 Figure. illustrates te fact tat te eistence of a it as : c never depends on ow te function ma or ma not be defined at c. Te function f as it as : even toug f is not defined at. Te function g as it as : even toug g() Z. Te function is te onl one wose it as : equals its value at = f() =, g() = (c) () = +, = Figure. : f() = : g() = : () =. Properties of Limits B appling si basic facts about its, we can calculate man unfamiliar its from its we alread know. For instance, from knowing tat and (k) = k Limit of te function wit constant value k () = c, Limit of te identit function at = c we can calculate te its of all polnomial and rational functions. Te facts are listed in Teorem. THEOREM Properties of Limits If L, M, c, and k are real numbers and f() = L and g() = M, ten. Sum Rule: (f() + g()) = L + M Te it of te sum of two functions is te sum of teir its.. Difference Rule: (f() - g()) = L - M Te it of te difference of two functions is te difference of teir its. continued

5 6 Capter Limits and Continuit 3. Product Rule: Te it of a product of two functions is te product of teir its. 4. Constant Multiple Rule: Te it of a constant times a function is te constant times te it of te function. 5. Quotient Rule: Te it of a quotient of two functions is te quotient of teir its, provided te it of te denominator is not zero. 6. Power Rule: If r and s are integers, s Z 0, ten provided tat is a real number. Te it of a rational power of a function is tat power of te it of te function, provided te latter is a real number. L r>s (f() # g()) = L # M (k # f()) = k # L f() g() = L M, M Z 0 (f())r>s = L r>s Here are some eamples of ow Teorem can be used to find its of polnomial and rational functions. Using Analtic Metods We remind te student tat unless oterwise stated all eamples and eercises are to be done using analtic algebraic metods witout te use of graping calculators or computer algebra sstems. EXAMPLE 3 Using Properties of Limits Use te observations k = k and = c, and te properties of its to find te following its. SOLUTION ( ) = = ( ) = = c 3 + 4c - 3 (4 + - ) ( + 5) = c4 + c - c Sum and Difference Rules Product and Constant Multiple Rules Quotient Rule Sum and Difference Rules Product Rule Now Tr Eercises 5 and 6. Eample 3 sows te remarkable strengt of Teorem. From te two simple observations tat k = k and = c, we can immediatel work our wa to its of polnomial functions and most rational functions using substitution.

6 Section. Rates of Cange and Limits 63 THEOREM Polnomial and Rational Functions. If f() = a n n + a n- n- + Á + a 0 is an polnomial function and c is an real number, ten f() = f(c) = a nc n + a n- c n- + Á + a 0.. If f() and g() are polnomials and c is an real number, ten f() g() = f(c), provided tat g(c) Z 0. g(c) EXAMPLE 4 Using Teorem :3 [ ( - )] = (3) ( - 3) = : + = () + () = 4 = 3 Now Tr Eercises 9 and. As wit polnomials, its of man familiar functions can be found b substitution at points were te are defined. Tis includes trigonometric functions, eponential and logaritmic functions, and composites of tese functions. Feel free to use tese properties. [ p, p] b [ 3, 3] Figure.3 Te grap of f() = (tan )> suggests tat f() : as : 0. (Eample 5) EXAMPLE 5 Using te Product Rule tan Determine. :0 SOLUTION Solve Analticall Using te analtic result of Eercise 77, we ave tan :0 = :0 a sin # cos b sin = :0 = # # :0 cos cos 0 = # =. tan = sin cos Product Rule Support Grapicall Te grap of f() = (tan )> in Figure.3 suggests tat te it eists and is about. Now Tr Eercise 33. Sometimes we can use a grap to discover tat its do not eist, as illustrated b Eample 6. EXAMPLE 6 Eploring a Noneistent Limit Use a grap to sow tat does not eist. 3 - : - continued

7 64 Capter Limits and Continuit SOLUTION Notice tat te denominator is 0 wen is replaced b, so we cannot use substitution to determine te it. Te grap in Figure.4 of f() = ( 3 - )>( - ) strongl suggests tat as : from eiter side, te absolute values of te function values get ver large. Tis, in turn, suggests tat te it does not eist. Now Tr Eercise 35. [ 0, 0] b [ 00, 00] Figure.4 Te grap of f() = ( 3 - )>( - ). (Eample 6) 4 = int One-Sided and Two-Sided Limits Sometimes te values of a function f tend to different its as approaces a number c from opposite sides. Wen tis appens, we call te it of f as approaces c from te rigt te rigt-and it of f at c and te it as approaces c from te left te left-and it of f at c. Here is te notation we use: rigt-and: left-and: f() + f() - Te it of f as approaces c from te rigt. Te it of f as approaces c from te left. EXAMPLE 7 Function Values Approac Two Numbers Te greatest integer function f() = int as different rigt-and and left-and its at eac integer, as we can see in Figure.5. For eample, :3 int = 3 and int =. + - Te it of int as approaces an integer n from te rigt is n, wile te it as approaces n from te left is n -. Now Tr Eercises 37 and 38. :3 Figure.5 At eac integer, te greatest integer function = int as different rigt-and and left-and its. (Eample 7) We sometimes call f() te two-sided it of f at c to distinguis it from te one-sided rigt-and and left-and its of f at c. Teorem 3 sows ow tese its are related. On te Far Side If f is not defined to te left of = c, ten f does not ave a left-and it at c. Similarl, if f is not defined to te rigt of = c, ten f does not ave a rigt-and it at c = f() Figure.6 Te grap of te function - +, 0 6, 6 f() = e, = -, , (Eample 8) THEOREM 3 One-Sided and Two-Sided Limits A function f() as a it as approaces c if and onl if te rigt-and and leftand its at c eist and are equal. In smbols, f() = L 3 f() = L and f() = L. + - Tus, te greatest integer function f() = int of Eample 7 does not ave a it as : 3 even toug eac one-sided it eists. EXAMPLE 8 Eploring Rigt- and Left-Hand Limits All te following statements about te function = f() graped in Figure.6 are true. At = 0 : f() =. :0 + At = : f() = 0 even toug f() =, : - f() =, + : f as no it as :. (Te rigt- and left-and its at are not equal, so : f() does not eist.) continued

8 Section. Rates of Cange and Limits 65 At = : f() =, : - f() =, + : f() = even toug f() =. : At = 3 : f() = f() = = f(3) = f(). - + :3 :3 :3 At = 4 : f() =. :4 - At noninteger values of c between 0 and 4, f as a it as : c. Now Tr Eercise 43. L g f Sandwic Teorem If we cannot find a it directl, we ma be able to find it indirectl wit te Sandwic Teorem. Te teorem refers to a function f wose values are sandwiced between te values of two oter functions, g and. If g and ave te same it as : c, ten f as tat it too, as suggested b Figure.7. O c Figure.7 Sandwicing f between g and forces te iting value of f to be between te iting values of g and. THEOREM 4 Te Sandwic Teorem If g() f() () for all Z c in some interval about c, and ten g() = () = L, f() = L. EXAMPLE 9 Using te Sandwic Teorem Sow tat. :0 [ sin(>)] = 0 SOLUTION We know tat te values of te sine function lie between - and. So, it follows tat and ` sin ` = ƒ ƒ # ` sin ` ƒ ƒ # = - sin. [ 0., 0.] b [ 0.0, 0.0] Figure.8 Te graps of =,, and 3 = - = sin (>). Notice tat 3. (Eample 9) Because, te Sandwic Teorem gives :0 (- ) = = 0 :0 :0 a sin b = 0. Te graps in Figure.8 support tis result.

9 66 Capter Limits and Continuit Quick Review. (For elp, go to Section..) Eercise numbers wit a gra background indicate problems tat te autors ave designed to be solved witout a calculator. In Eercises 4, find f().. f() = f() = f() = sin ap b 4. f() = c 3 -, 6 -, Ú In Eercises 5 8, write te inequalit in te form a 6 6 b. 5. ƒ ƒ ƒ ƒ 6 c 7. ƒ - ƒ ƒ - c ƒ 6 d In Eercises 9 and 0, write te fraction in reduced form Section. Eercises In Eercises 4, an object dropped from rest from te top of a tall building falls = 6t feet in te first t seconds.. Find te average speed during te first 3 seconds of fall.. Find te average speed during te first 4 seconds of fall. 3. Find te speed of te object at t = 3 seconds and confirm our answer algebraicall. 4. Find te speed of te object at t = 4 seconds and confirm our answer algebraicall. In Eercises 5 and 6, use k = k, = c, and te properties of its to find te it. 5. ( ) In Eercises 7 4, determine te it b substitution. Support grapicall. 7. :-> 3 ( - ) 8. ( + 3)998 :-4 9. : ( ) 0. : : ( - 6)>3 4. :-. :> int : + 3 In Eercises 5 0, complete te following tables and state wat ou believe :0 f() to be f()???? f()???? 5. f() = f() = - 7. f() = sin 8. f() = sin 9. f() = 0-0. f() = sin (ln ƒ ƒ ) In Eercises 4, eplain w ou cannot use substitution to determine te it. Find te it if it eists.. -. :- :0 ƒ ƒ (4 + ) :0 :0 In Eercises 5 34, determine te it grapicall. Confirm algebraicall. - t - 3t : - t: t : ( + ) :0 30. sin 3. :0-3. sin 33. :0 34. In Eercises 35 and 36, use a grap to sow tat te it does not eist : - In Eercises 37 4, determine te it. :0 sin :0 + sin :0 3-5 :5-5 : int 38. int :0 + :0 -

10 Section. Rates of Cange and Limits int 40. int 47. :0.0 : ƒ ƒ :0 + :0 - In Eercises 43 and 44, wic of te statements are true about te function = f() graped tere, and wic are false? 43. = f() 0 :- + :0 - (c) :0 - (d) f() = - :0 :0 + (e) f() eists :0 (f) :0 (g) :0 () : (i) : ( j) : f() ƒ ƒ 4 = f() = p(s) = F() s (c) :0 - f() :0 + f() f() :0 (d) f(0) 48. p(s) s:- - (c) s:- + p(s) p(s) s:- (d) p(-) 49. F() :0 - :0 + F() (c) F() :0 (d) F(0) 0 3 :- + does not eist. : (c) : (d) : - (e) : + (f ) does not eist. : (g) f() = + :0 :0 - () eists at ever c in (-, ). (i) eists at ever c in (, 3). In Eercises 45 50, use te grap to estimate te its and value of te function, or eplain w te its do not eist. 45. :3-4 = f() 3 = g(t) t (c) :3 + f() f() :3 (d) f(3) 46. g(t) t:-4 - g(t) t:-4 + (c) g(t) t:-4 (d) g(-4) 50. G() : - G() = G() : + (c) G() : (d) G() In Eercises 5 54, matc te function wit te table = - - = = + - = X X =.7 X X =.7 Y Y ERROR (c) X X =.7 X X =.7 Y ERROR Y.3.. ERROR...3 (d)

11 68 Capter Limits and Continuit In Eercises 55 and 56, determine te it. 55. Assume tat f() = 0 and g() = 3. :4 :4 (g() + 3) f() :4 :4 g() (c) (d) :4 g () :4 f() Assume tat f() = 7 and g() = -3. :b :b (f() + g()) (f() # g()) :b :b f() (c) 4 g() (d) :b :b g() In Eercises 57 60, complete parts,, and (c) for te piecewise-defined function. Draw te grap of f. Determine + f() and - f(). (c) Writing to Learn Does f() eist? If so, wat is it? If not, eplain , 6 c =, f() = c +, > 3 -, < c =, f() = c, = >, > -, < 59. c =, f() = c , Ú 60. c = -, f() = b -, Z -, = - In Eercises 6 64, complete parts (d) for te piecewise-defined function. Draw te grap of f. At wat points c in te domain of f does f() eist? (c) At wat points c does onl te left-and it eist? (d) At wat points c does onl te rigt-and it eist? sin, -p f() = b cos, 0 p cos, -p f() = b sec, 0 p -, f() = c, 6, =, - 6 0, or f() = c, = 0 0, 6 -, or 7 In Eercises 65 68, find te it grapicall. Use te Sandwic Teorem to confirm our answer. 65. sin 66. :0 sin : :0 cos :0 sin 69. Free Fall A water balloon dropped from a window ig above te ground falls = 4.9t m in t sec. Find te balloon s average speed during te first 3 sec of fall. speed at te instant t = Free Fall on a Small Airless Planet A rock released from rest to fall on a small airless planet falls = gt m in t sec, g a constant. Suppose tat te rock falls to te bottom of a crevasse 0 m below and reaces te bottom in 4 sec. Find te value of g. Find te average speed for te fall. (c) Wit wat speed did te rock it te bottom? Standardized Test Questions 7. True or False If f() = and f() =, ten f() = - +. Justif our answer. + sin 7. True or False =. Justif our answer. :0 In Eercises 73 76, use te following function. -, f() = c +, Multiple Coice Wat is te value of : - f()? (A) 5> (B) 3> (C) (D) 0 (E) does not eist 74. Multiple Coice Wat is te value of : + f()? (A) 5> (B) 3> (C) (D) 0 (E) does not eist 75. Multiple Coice Wat is te value of : f()? (A) 5> (B) 3> (C) (D) 0 (E) does not eist 76. Multiple Coice Wat is te value of f()? (A) 5> (B) 3> (C) (D) 0 (E) does not eist 77. Group Activit To prove tat u:0 (sin u)>u = wen u is measured in radians, te plan is to sow tat te rigt- and leftand its are bot. To sow tat te rigt-and it is, eplain w we can restrict our attention to 0 6 u 6 p>. Use te figure to sow tat area of OAP = sin u, area of sector OAP = u, area of OAT = tan u.

12 Section. Rates of Cange and Limits 69 O cos sin (c) Use part and te figure to sow tat for 0 6 u 6 p>, Q. sin u 6 u 6 tan u tan A(, 0) (d) Sow tat for 0 6 u 6 p> te inequalit of part (c) can be written in te form P 6 u sin u 6 cos u. (e) Sow tat for 0 6 u 6 p> te inequalit of part (d) can be written in te form cos u 6 sin u u 6. (f) Use te Sandwic Teorem to sow tat sin u =. u:0 + u T (g) Sow tat (sin u)>u is an even function. () Use part (g) to sow tat (i) Finall, sow tat Etending te Ideas 78. Controlling Outputs Let f() = 3 -. Sow tat : f() = = f(). Use a grap to estimate values for a and b so tat.8 6 f() 6. provided a 6 6 b. (c) Use a grap to estimate values for a and b so tat.99 6 f() 6.0 provided a 6 6 b. 79. Controlling Outputs Let f() = sin. Find f(p>6). sin u =. u:0 - u sin u =. u:0 u Use a grap to estimate an interval (a, b) about = p>6 so tat f() provided a 6 6 b. (c) Use a grap to estimate an interval (a, b) about = p>6 so tat f() provided a 6 6 b. 80. Limits and Geometr Let P(a, a ) be a point on te parabola =, a 7 0. Let O be te origin and (0, b) te -intercept of te perpendicular bisector of line segment OP. Find P:O b.