60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can fail to eist. Stud and use a formal definition of it. f() = (, ) An Introduction to Limits Suppose ou are asked to sketch the graph of the function f given b f,. For all values other than, ou can use standard curve-sketching techniques. However, at, it is not clear what to epect. To get an idea of the behavior of the graph of f near, ou can use two sets of -values one set that approaches from the left and one set that approaches from the right, as shown in the table. approaches from the left. approaches from the right. f 0.75 0.9 0.99 0.999.00.0..5..70.970.997?.00.00.0.8 f() = f approaches. f approaches. The it of f as approaches is. Figure.5 The graph of f is a parabola that has a gap at the point,, as shown in Figure.5. Although cannot equal, ou can move arbitraril close to, and as a result f moves arbitraril close to. Using it notation, ou can write f. This is read as the it of f as approaches is. This discussion leads to an informal description of a it. If f becomes arbitraril close to a single number L as approaches c from either side, the it of f, as approaches c, is L. This it is written as f L. c EXPLORATION The discussion above gives an eample of how ou can estimate a it numericall b constructing a table and graphicall b drawing a graph. Estimate the following it numericall b completing the table. f.75.9.99.999.00.0..5????????? Then use a graphing utilit to estimate the it graphicall.
60_00.qd //0 :05 PM Page 9 SECTION. Finding Limits Graphicall and Numericall 9 f is undefined at = 0. EXAMPLE Estimating a Limit Numericall Evaluate the function at several points near 0 and use the results to estimate the it 0. f Solution The table lists the values of f for several -values near 0. f() = + approaches 0 from the left. approaches 0 from the right. 0.0 0.00 0.000 0 0.000 0.00 0.0 f.9999.99950.99995?.00005.00050.0099 f approaches. f approaches. The it of f as approaches 0 is. Figure.6 From the results shown in the table, ou can estimate the it to be. This it is reinforced b the graph of f (see Figure.6). In Eample, note that the function is undefined at 0 and et f () appears to be approaching a it as approaches 0. This often happens, and it is important to realize that the eistence or noneistence of f at c has no bearing on the eistence of the it of f as approaches c. EXAMPLE Finding a Limit Find the it of f as approaches where f is defined as f, 0,., f() = 0, The it of f as approaches is. Figure.7 Solution Because f for all other than, ou can conclude that the it is, as shown in Figure.7. So, ou can write f. The fact that f 0 has no bearing on the eistence or value of the it as approaches. For instance, if the function were defined as f,, the it would be the same. So far in this section, ou have been estimating its numericall and graphicall. Each of these approaches produces an estimate of the it. In Section., ou will stud analtic techniques for evaluating its. Throughout the course, tr to develop a habit of using this three-pronged approach to problem solving.. Numerical approach Construct a table of values.. Graphical approach Draw a graph b hand or using technolog.. Analtic approach Use algebra or calculus.
60_00.qd //0 :05 PM Page 50 50 CHAPTER Limits and Their Properties Limits That Fail to Eist In the net three eamples ou will eamine some its that fail to eist. EXAMPLE Behavior That Differs from the Right and Left δ δ f() = f does not eist. 0 Figure.8 f() = f() = Show that the it does not eist. 0 Solution Consider the graph of the function From Figure.8, ou can see that for positive -values, and for negative -values, This means that no matter how close gets to 0, there will be both positive and negative -values that ield f and f. Specificall, if (the lowercase Greek letter delta) is a positive number, then for -values satisfing the inequalit 0 < <, ou can classif the values of as shown., 0 > 0 < 0. 0, f. Negative -values Positive -values.. ield ield This implies that the it does not eist. EXAMPLE Unbounded Behavior Discuss the eistence of the it 0. f() = Solution Let f. In Figure.9, ou can see that as approaches 0 from either the right or the left, f increases without bound. This means that b choosing close enough to 0, ou can force f to be as large as ou want. For instance, f ) will be larger than 00 if ou choose that is within of 0. That is, 0 < < 0 f > 00. 0 Similarl, ou can force f to be larger than,000,000, as follows. f does not eist. 0 Figure.9 0 < < 000 f >,000,000 Because f is not approaching a real number L as approaches 0, ou can conclude that the it does not eist.
60_00.qd //0 :05 PM Page 5 SECTION. Finding Limits Graphicall and Numericall 5 EXAMPLE 5 Oscillating Behavior f() = sin Discuss the eistence of the it Solution Let f sin. In Figure.0, ou can see that as approaches 0, f oscillates between and. So, the it does not eist because no matter how small ou choose, it is possible to choose and within units of 0 such that sin and sin, as shown in the table. 5 sin 0. 7 9 0 sin / Limit does not eist. f does not eist. 0 Figure.0 Common Tpes of Behavior Associated with Noneistence of a Limit. f approaches a different number from the right side of c than it approaches from the left side.. f increases or decreases without bound as approaches c.. f oscillates between two fied values as approaches c. There are man other interesting functions that have unusual it behavior. An often cited one is the Dirichlet function f 0,, if is rational. if is irrational. Because this function has no it at an real number c, it is not continuous at an real number c. You will stud continuit more closel in Section.. TECHNOLOGY PITFALL When ou use a graphing utilit to investigate the behavior of a function near the -value at which ou are tring to evaluate a it, remember that ou can t alwas trust the pictures that graphing utilities draw. If ou use a graphing utilit to graph the function in Eample 5 over an interval containing 0, ou will most likel obtain an incorrect graph such as that shown in Figure.. The reason that a graphing utilit can t show the correct graph is that the graph has infinitel man oscillations over an interval that contains 0. The Granger Collection 0.5. 0.5 PETER GUSTAV DIRICHLET (805 859) In the earl development of calculus, the definition of a function was much more restricted than it is toda, and functions such as the Dirichlet function would not have been considered. The modern definition of function was given b the German mathematician Peter Gustav Dirichlet. Incorrect graph of Figure.. f sin. indicates that in the HM mathspace CD-ROM and the online Eduspace sstem for this tet, ou will find an Open Eploration, which further eplores this eample using the computer algebra sstems Maple, Mathcad, Mathematica, and Derive.
60_00.qd //0 :05 PM Page 5 5 CHAPTER Limits and Their Properties L + ε L L ε (c, L) c + δ c c δ The - definition of the it of f as approaches c Figure. A Formal Definition of Limit Let s take another look at the informal description of a it. If f becomes arbitraril close to a single number L as approaches c from either side, then the it of f as approaches c is L, written as f L. c At first glance, this description looks fairl technical. Even so, it is informal because eact meanings have not et been given to the two phrases f becomes arbitraril close to L and approaches c. The first person to assign mathematicall rigorous meanings to these two phrases was Augustin-Louis Cauch. His - definition of it is the standard used toda. In Figure., let (the lowercase Greek letter epsilon) represent a (small) positive number. Then the phrase f becomes arbitraril close to L means that f lies in the interval L, L. Using absolute value, ou can write this as f L <. Similarl, the phrase approaches c means that there eists a positive number such that lies in either the interval c, c or the interval c, c. This fact can be concisel epressed b the double inequalit 0 < c < The first inequalit 0 < c The distance between and c is more than 0. epresses the fact that c. The second inequalit c < is within sas that is within a distance. units of c. of c. Definition of Limit Let f be a function defined on an open interval containing c (ecept possibl at c) and let L be a real number. The statement f L c means that for each > 0 there eists a > 0 such that if 0 < c <, then f L <. FOR FURTHER INFORMATION For more on the introduction of rigor to calculus, see Who Gave You the Epsilon? Cauch and the Origins of Rigorous Calculus b Judith V. Grabiner in The American Mathematical Monthl. To view this article, go to the website www.matharticles.com. NOTE Throughout this tet, the epression f L c implies two statements the it eists and the it is L. Some functions do not have its as c, but those that do cannot have two different its as c. That is, if the it of a function eists, it is unique (see Eercise 69).
60_00.qd //0 :05 PM Page 5 SECTION. Finding Limits Graphicall and Numericall 5 - The net three eamples should help ou develop a better understanding of the definition of it. =.0 = = 0.99 =.995 = =.005 f() = 5 The it of f as approaches is. Figure. EXAMPLE 6 Finding a for a Given Given the it 5 find such that 5 < 0.0 whenever Solution In this problem, ou are working with a given value of namel, 0.0. To find an appropriate, notice that 5 6. Because the inequalit ou can choose 0 < < 0.005 implies that 5 as shown in Figure.. 5 < 0.0 is equivalent to This choice works because 0.0 0.005. < 0.005 0.0 0 < <. < 0.0, NOTE In Eample 6, note that 0.005 is the largest value of that will guarantee 5 < 0.0 whenever 0 < <. An smaller positive value of would also work. In Eample 6, ou found a -value for a given. This does not prove the eistence of the it. To do that, ou must prove that ou can find a for an, as shown in the net eample. = + ε = = ε = + δ = = δ f() = The it of f as approaches is. Figure. EXAMPLE 7 Using the - Definition of Limit Use the - definition of it to prove that. Solution You must show that for each > 0, there eists a > 0 such that < whenever 0 < <. Because our choice of depends on, ou need to establish a connection between the absolute values and. 6 So, for a given > 0 ou can choose This choice works because 0 < < implies that < as shown in Figure...
60_00.qd //0 :05 PM Page 5 5 CHAPTER Limits and Their Properties EXAMPLE 8 Using the - Definition of Limit f() = + ε ( + δ) ( δ) ε + δ δ The it of f as approaches is. Figure.5 Use the -. definition of it to prove that Solution You must show that for each > 0, there eists a > 0 such that < whenever 0 < <. To find an appropriate, begin b writing For all in the interval ou know that.,, So, letting be the minimum of 5 and, it follows that, whenever 0 < < 5. <, ou have < 5 5 as shown in Figure.5. Throughout this chapter ou will use the definition of it primaril to prove theorems about its and to establish the eistence or noneistence of particular tpes of its. For finding its, ou will learn techniques that are easier to use than the definition of it. - - Eercises for Section. In Eercises 8, complete the table and use the result to estimate the it. Use a graphing utilit to graph the function to confirm our result. 5. See www.calcchat.com for worked-out solutions to odd-numbered eercises.. f.9.99.999.00.0. f.9.99.999.00.0. 6. 5. f.9.99.999.00.0. f.9.99.999.00.0. 7. sin 0. 0 f 0. 0.0 0.00 0.00 0.0 0.. f 0. 0.0 0.00 0.00 0.0 0. cos 8. 0 f 0. 0.0 0.00 0.00 0.0 0...0.00.999.99.9 f
60_00.qd //0 :05 PM Page 55 SECTION. Finding Limits Graphicall and Numericall 55 In Eercises 9 8, use the graph to find the it (if it eists). If the it does not eist, eplain wh. 9. 0.. f... 5 5 5. sin 6. f, 0, 5 6 789 f sec 0 f,, In Eercises 9 and 0, use the graph of the function f to decide whether the value of the given quantit eists. If it does, find it. If not, eplain wh. 9. (a) f (b) (c) f (d) 0. (a) f (b) f (c) (d) (e) f 0 f 0 f 5 (f ) f (g) (h) f f In Eercises and, use the graph of f to identif the values of c for which f eists... f f 6 6 c 6 6 5 5 6 7. cos 8. 0 π tan π π π π π In Eercises and, sketch the graph of f. Then identif the values of c for which f eists.. f, 8,, c sin,. f cos, cos, < < < 0 0 >
60_00.qd //0 :06 PM Page 56 56 CHAPTER Limits and Their Properties In Eercises 5 and 6, sketch a graph of a function f that satisfies the given values. (There are man correct answers.) 5. f 0 is undefined. 6. f 0 f 0 f 6 f f does not eist. 7. Modeling Data The cost of a telephone call between two cities is $0.75 for the first minute and $0.50 for each additional minute or fraction thereof. A formula for the cost is given b Ct 0.75 0.50t where t is the time in minutes. Note: greatest integer n such that n. For eample,. and.6. (a) Use a graphing utilit to graph the cost function for 0 < t 5. (b) Use the graph to complete the table and observe the behavior of the function as t approaches.5. Use the graph and the table to find C t. t.5 t C f 0 f 0...5.6.7? (c) Use the graph to complete the table and observe the behavior of the function as t approaches. 0. The graph of f is shown in the figure. Find such that if f < 0.0. 0 < <.0.5.0 0.5. The graph of f is shown in the figure. Find such that if f < 0.. 0 < < =. = = 0.9. The graph of f f.0.00 0.99 0 0 99 99 then then t C.5.9..5? is shown in the figure. Find such that if f < 0.. 0 < < then Does the it of Ct as t approaches eist? Eplain. 8. Repeat Eercise 7 for Ct 0.5 0.t. 9. The graph of f is shown in the figure. Find such that if then 0 < < f < 0.. 5..6 0.5.0.5.0.5.0.6. In Eercises 6, find the it L. Then find > 0 such that whenever 0 < c <. f L < 0.0.. f =. = =.8 5. 6. 5 The smbol indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. The solutions of other eercises ma also be facilitated b use of appropriate technolog.
60_00.qd //0 :06 PM Page 57 SECTION. Finding Limits Graphicall and Numericall 57 In Eercises 7 8, find the it Then use the - definition to prove that the it is L. 7. 8. 9. 0..... 5. 6. 7. 8. Writing In Eercises 9 5, use a graphing utilit to graph the function and estimate the it (if it eists). What is the domain of the function? Can ou detect a possible error in determining the domain of a function solel b analzing the graph generated b a graphing utilit? Write a short paragraph about the importance of eamining a function analticall as well as graphicall. 9. 50. 5. 5. 9 6 0 f f ) f f f 9 f 9 f 9 f 5 5 Writing About Concepts 5. Write a brief description of the meaning of the notation f 5. 8 5. If f, can ou conclude anthing about the it of f as approaches? Eplain our reasoning. 55. If the it of f as approaches is, can ou conclude anthing about f? Eplain our reasoning. L. Writing About Concepts (continued) 56. Identif three tpes of behavior associated with the noneistence of a it. Illustrate each tpe with a graph of a function. 57. Jewelr A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) If the ring s inner circumference can var between 5.5 centimeters and 6.5 centimeters, how can the radius var? (c) Use the - definition of it to describe this situation. Identif and. 58. Sports A sporting goods manufacturer designs a golf ball having a volume of.8 cubic inches. (a) What is the radius of the golf ball? (b) If the ball s volume can var between.5 cubic inches and.5 cubic inches, how can the radius var? (c) Use the - definition of it to describe this situation. Identif and. 59. Consider the function f. Estimate the it 0 b evaluating f at -values near 0. Sketch the graph of f. 60. Consider the function f. Estimate 0 b evaluating f at -values near 0. Sketch the graph of f. 6. Graphical Analsis The statement means that for each > 0 there corresponds a > 0 such that if 0 < <, then <. If 0.00, then < 0.00. Use a graphing utilit to graph each side of this inequalit. Use the zoom feature to find an interval, such that the graph of the left side is below the graph of the right side of the inequalit.
60_00.qd //0 :06 PM Page 58 58 CHAPTER Limits and Their Properties 6. Graphical Analsis The statement means that for each > 0 there corresponds a > 0 such that if, then 0 < < <. If 0.00, then < 0.00. Use a graphing utilit to graph each side of this inequalit. Use the zoom feature to find an interval, such that the graph of the left side is below the graph of the right side of the inequalit. True or False? In Eercises 6 66, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 6. If f is undefined at c, then the it of f as approaches c does not eist. 6. If the it of f as approaches c is 0, then there must eist a number k such that f k < 0.00. 65. If f c L, then f L. c 66. If f L, then f c L. c 67. Consider the function f. (a) Is 0.5 a true statement? Eplain. (b) Is 0.5 0 0 a true statement? Eplain. 68. Writing The definition of it on page 5 requires that f is a function defined on an open interval containing c, ecept possibl at c. Wh is this requirement necessar? 69. Prove that if the it of f as c eists, then the it must be unique. [Hint: Let f L c and f L c and prove that L L.] 70. Consider the line f m b, where m 0. Use the definition of it to prove that f mc b. c 7. Prove that f L is equivalent to f L 0. c c - 7. (a) Given that prove that there eists an open interval a, b containing 0 such that 0.0 > 0 for all 0 in a, b. (b) Given that g L, where L > 0, prove that there c eists an open interval a, b containing c such that g > 0 for all c in a, b. 7. Programming Use the programming capabilities of a graphing utilit to write a program for approimating f. c Assume the program will be applied onl to functions whose its eist as approaches c. Let f and generate two lists whose entries form the ordered pairs c ± 0. n, f c ± 0. n for n 0,,,, and. 7. Programming Use the program ou created in Eercise 7 to approimate the it 0.0 0.0 0. Putnam Eam Challenge 75. Inscribe a rectangle of base b and height h and an isosceles triangle of base b in a circle of radius one as shown. For what value of h do the rectangle and triangle have the same area? h b 76. A right circular cone has base of radius and height. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? These problems were composed b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved.