Limits and Their Properties

Transcription

1 60_000.qd //0 :8 PM Page Cost Utilit companies use a platinum catalst pollution scrubber to remove pollutants from smokestack emissions. Which do ou think is more costl for a utilit compan to accomplish removing the first 90 percent of the pollutants, or removing the last 0 percent? Wh? Cost Percent removed Limits and Their Properties Cost Percent removed Percent removed The it process is a fundamental concept of calculus. One technique ou can use to estimate a it is to graph the function and then determine the behavior of the graph as the independent variable approaches a specific value. In Chapter, ou will learn how to find its of functions analticall, graphicall, and numericall. Jerem Walker/Gett Images

2 60_00.qd //0 : PM Page CHAPTER Limits and Their Properties The Mistress Fellows, Girton College, Cambridge Section. STUDY TIP As ou progress through this course, remember that learning calculus is just one of our goals. Your most important goal is to learn how to use calculus to model and solve real-life problems. Here are a few problemsolving strategies that ma help ou. Be sure ou understand the question. What is given? What are ou asked to find? Outline a plan. There are man approaches ou could use: look for a pattern, solve a simpler problem, work backwards, draw a diagram, use technolog, or an of man other approaches. Complete our plan. Be sure to answer the question. Verbalize our answer. For eample, rather than writing the answer as.6, it would be better to write the answer as The area of the region is.6 square meters. Look back at our work. Does our answer make sense? Is there a wa ou can check the reasonableness of our answer? GRACE CHISHOLM YOUNG (868 9) Grace Chisholm Young received her degree in mathematics from Girton College in Cambridge, England. Her earl work was published under the name of William Young, her husband. Between 9 and 96, Grace Young published work on the foundations of calculus that won her the Gamble Prize from Girton College. A Preview of Calculus Understand what calculus is and how it compares with precalculus. Understand that the tangent line problem is basic to calculus. Understand that the area problem is also basic to calculus. What Is Calculus? Calculus is the mathematics of change velocities and accelerations. Calculus is also the mathematics of tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variet of other concepts that have enabled scientists, engineers, and economists to model real-life situations. Although precalculus mathematics also deals with velocities, accelerations, tangent lines, slopes, and so on, there is a fundamental difference between precalculus mathematics and calculus. Precalculus mathematics is more static, whereas calculus is more dnamic. Here are some eamples. An object traveling at a constant velocit can be analzed with precalculus mathematics. To analze the velocit of an accelerating object, ou need calculus. The slope of a line can be analzed with precalculus mathematics. To analze the slope of a curve, ou need calculus. A tangent line to a circle can be analzed with precalculus mathematics. To analze a tangent line to a general graph, ou need calculus. The area of a rectangle can be analzed with precalculus mathematics. To analze the area under a general curve, ou need calculus. Each of these situations involves the same general strateg the reformulation of precalculus mathematics through the use of a it process. So, one wa to answer the question What is calculus? is to sa that calculus is a it machine that involves three stages. The first stage is precalculus mathematics, such as the slope of a line or the area of a rectangle. The second stage is the it process, and the third stage is a new calculus formulation, such as a derivative or integral. Precalculus mathematics Limit process Calculus Some students tr to learn calculus as if it were simpl a collection of new formulas. This is unfortunate. If ou reduce calculus to the memorization of differentiation and integration formulas, ou will miss a great deal of understanding, self-confidence, and satisfaction. On the following two pages some familiar precalculus concepts coupled with their calculus counterparts are listed. Throughout the tet, our goal should be to learn how precalculus formulas and techniques are used as building blocks to produce the more general calculus formulas and techniques. Don t worr if ou are unfamiliar with some of the old formulas listed on the following two pages ou will be reviewing all of them. As ou proceed through this tet, come back to this discussion repeatedl. Tr to keep track of where ou are relative to the three stages involved in the stud of calculus. For eample, the first three chapters break down as shown. Chapter P: Preparation for Calculus Precalculus Chapter : Limits and Their Properties Limit process Chapter : Differentiation Calculus

3 60_00.qd //0 : PM Page SECTION. A Preview of Calculus Without Calculus With Differential Calculus Value of f = f() Limit of f as when c approaches c c c = f() Slope of a line Slope of a curve d d Secant line to a curve Tangent line to a curve Average rate of change between t a and t b Instantaneous t = a t = b rate of change at t c t = c Curvature of a circle Curvature of a curve Height of a curve when c c Maimum height of a curve on an interval a b Tangent plane to a sphere Tangent plane to a surface Direction of motion along a line Direction of motion along a curve

4 60_00.qd //0 : PM Page CHAPTER Limits and Their Properties Without Calculus With Integral Calculus Area of a rectangle Area under a curve Work done b a constant force Work done b a variable force Center of a rectangle Centroid of a region Length of a line segment Length of an arc Surface area of a clinder Surface area of a solid of revolution Mass of a solid of constant densit Mass of a solid of variable densit Volume of a rectangular solid Volume of a region under a surface Sum of a finite number of terms a a... a n S Sum of an infinite number of terms a a a... S

5 60_00.qd //0 : PM Page 5 SECTION. A Preview of Calculus 5 = f() Tangent line P The tangent line to the graph of f at P Figure. The Tangent Line Problem The notion of a it is fundamental to the stud of calculus. The following brief descriptions of two classic problems in calculus the tangent line problem and the area problem should give ou some idea of the wa its are used in calculus. In the tangent line problem, ou are given a function f and a point P on its graph and are asked to find an equation of the tangent line to the graph at point P, as shown in Figure.. Ecept for cases involving a vertical tangent line, the problem of finding the tangent line at a point P is equivalent to finding the slope of the tangent line at P. You can approimate this slope b using a line through the point of tangenc and a second point on the curve, as shown in Figure.(a). Such a line is called a secant line. If Pc, f c is the point of tangenc and Qc, fc is a second point on the graph of points is given b f, the slope of the secant line through these two m sec f c f c c c f c f c. Q(c +, f(c + )) P(c, f(c)) f(c + ) f(c) Q Secant lines P Tangent line (a) The secant line through c, f c and c, fc Figure. (b) As Q approaches P, the secant lines approach the tangent line. As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line, as shown in Figure.(b). When such a iting position eists, the slope of the tangent line is said to be the it of the slope of the secant line. (Much more will be said about this important problem in Chapter.) EXPLORATION The following points lie on the graph of f. Q.5, f.5, Q., f., Q.0, f.0, Q.00, f.00, Q 5.000, f.000 Each successive point gets closer to the point P,. Find the slope of the secant line through Q and P, Q and P, and so on. Graph these secant lines on a graphing utilit. Then use our results to estimate the slope of the tangent line to the graph of f at the point P.

6 60_00.qd //0 : PM Page 6 6 CHAPTER Limits and Their Properties a Area under a curve Figure. = f() b The Area Problem In the tangent line problem, ou saw how the it process can be applied to the slope of a line to find the slope of a general curve. A second classic problem in calculus is finding the area of a plane region that is bounded b the graphs of functions. This problem can also be solved with a it process. In this case, the it process is applied to the area of a rectangle to find the area of a general region. As a simple eample, consider the region bounded b the graph of the function f, the -ais, and the vertical lines a and b, as shown in Figure.. You can approimate the area of the region with several rectangular regions, as shown in Figure.. As ou increase the number of rectangles, the approimation tends to become better and better because the amount of area missed b the rectangles decreases. Your goal is to determine the it of the sum of the areas of the rectangles as the number of rectangles increases without bound. = f() = f() HISTORICAL NOTE In one of the most astounding events ever to occur in mathematics, it was discovered that the tangent line problem and the area problem are closel related. This discover led to the birth of calculus. You will learn about the relationship between these two problems when ou stud the Fundamental Theorem of Calculus in Chapter. a b a b Approimation using four rectangles Figure. Approimation using eight rectangles EXPLORATION Consider the region bounded b the graphs of f, 0, and, as shown in part (a) of the figure. The area of the region can be approimated b two sets of rectangles one set inscribed within the region and the other set circumscribed over the region, as shown in parts (b) and (c). Find the sum of the areas of each set of rectangles. Then use our results to approimate the area of the region. f() = f() = f() = (a) Bounded region (b) Inscribed rectangles (c) Circumscribed rectangles

7 60_00.qd //0 : PM Page 7 SECTION. A Preview of Calculus 7 Eercises for Section. In Eercises 6, decide whether the problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, eplain our reasoning and use a graphical or numerical approach to estimate the solution.. Find the distance traveled in 5 seconds b an object traveling at a constant velocit of 0 feet per second.. Find the distance traveled in 5 seconds b an object moving with a velocit of vt 0 7 cos t feet per second.. A bicclist is riding on a path modeled b the function f 0.08, where and f are measured in miles. Find the rate of change of elevation when. Figure for Figure for. A bicclist is riding on a path modeled b the function f 0.08, where and f are measured in miles. Find the rate of change of elevation when. 5. Find the area of the shaded region. 5 (, ) (0, 0) ( ) f() = (5, 0) 5 6 Figure for 5 Figure for 6 6. Find the area of the shaded region. 7. Secant Lines Consider the function f and the point P, on the graph of f. (a) Graph f and the secant lines passing through P, and Q, f for -values of,.5, and 0.5. (b) Find the slope of each secant line. (c) Use the results of part (b) to estimate the slope of the tangent line of f at P,. Describe how to improve our approimation of the slope. 8. Secant Lines Consider the function f and the point P, on the graph of f. (a) Graph f and the secant lines passing through P, and Q, f for -values of,, and 5. (b) Find the slope of each secant line. f() = See for worked-out solutions to odd-numbered eercises. (c) Use the results of part (b) to estimate the slope of the tangent line of f at P,. Describe how to improve our approimation of the slope. 9. (a) Use the rectangles in each graph to approimate the area of the region bounded b 5, 0,, and 5. 5 (b) Describe how ou could continue this process to obtain a more accurate approimation of the area. 0. (a) Use the rectangles in each graph to approimate the area of the region bounded b sin, 0, 0, and. π 5 π (b) Describe how ou could continue this process to obtain a more accurate approimation of the area. Writing About Concepts. Consider the length of the graph of f 5 from, 5 to 5,. 5 (, 5) (5, ) 5 (a) Approimate the length of the curve b finding the distance between its two endpoints, as shown in the first figure. (b) Approimate the length of the curve b finding the sum of the lengths of four line segments, as shown in the second figure. (c) Describe how ou could continue this process to obtain a more accurate approimation of the length of the curve. 5 (, 5) 5 π 5 π 5 (5, )

8 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 ? 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 ????????? Then use a graphing utilit to estimate the it graphicall.

9 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 f ? 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.

10 60_00.qd //0 :05 PM Page 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 > 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.

11 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 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 PETER GUSTAV DIRICHLET ( ) 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.

12 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 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).

13 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 < < implies that 5 as shown in Figure.. 5 < 0.0 is equivalent to This choice works because < < <. < 0.0, NOTE In Eample 6, note that 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...

14 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 for worked-out solutions to odd-numbered eercises.. f f f f sin 0. 0 f f cos 8. 0 f f

15 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 f sin 6. f, 0, 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 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 >

16 60_00.qd //0 :06 PM Page 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 t 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 ? (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 < < < The graph of f is shown in the figure. Find such that if f < < < =. = = 0.9. The graph of f f then then t C ? is shown in the figure. Find such that if f < < < then Does the it of Ct as t approaches eist? Eplain. 8. Repeat Eercise 7 for Ct t. 9. The graph of f is shown in the figure. Find such that if then 0 < < f < In Eercises 6, find the it L. Then find > 0 such that whenever 0 < c <. f L < f =. = = 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.

17 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 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 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 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 < 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.

18 60_00.qd //0 :06 PM Page 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 < 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 < 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 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 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.

19 60_00.qd //0 : PM Page 59 SECTION. Evaluating Limits Analticall 59 Section. Evaluating Limits Analticall Evaluate a it using properties of its. Develop and use a strateg for finding its. Evaluate a it using dividing out and rationalizing techniques. Evaluate a it using the Squeeze Theorem. Properties of Limits In Section., ou learned that the it of f as approaches c does not depend on the value of f at c. It ma happen, however, that the it is precisel fc. In such cases, the it can be evaluated b direct substitution. That is, f fc. c Substitute c for. Such well-behaved functions are continuous at c. You will eamine this concept more closel in Section.. f(c) = THEOREM. Some Basic Limits c + ε f(c) = c ε = δ Let b and c be real numbers and let n be a positive integer.. b b. c. c n c n c c ε = δ c ε c δ Figure.6 c c + δ Proof To prove Propert of Theorem., ou need to show that for each > 0 there eists a > 0 such that c < whenever 0 < c <. To do this, choose The second inequalit then implies the first, as shown in Figure.6. This completes the proof. (Proofs of the other properties of its in this section are listed in Appendi A or are discussed in the eercises.). NOTE When ou encounter new notations or smbols in mathematics, be sure ou know how the notations are read. For instance, the it in Eample (c) is read as the it of as approaches is. EXAMPLE Evaluating Basic Limits a. b. c. THEOREM. Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following its. f L c and g K c. Scalar multiple:. Sum or difference:. Product: bf bl c f ± g L ± K c fg LK c. Quotient: f c g L K, provided K 0 5. Power: f n L n c

20 60_00.qd //0 : PM Page CHAPTER Limits and Their Properties EXAMPLE The Limit of a Polnomial 9 Propert Propert Eample Simplif. In Eample, note that the it (as ) of the polnomial function p is simpl the value of p at. p p 9 This direct substitution propert is valid for all polnomial and rational functions with nonzero denominators. THEOREM. Limits of Polnomial and Rational Functions If p is a polnomial function and c is a real number, then p pc. c If r is a rational function given b r pq and c is a real number such that qc 0, then pc r rc c qc. EXAMPLE The Limit of a Rational Function Find the it:. Solution Because the denominator is not 0 when, ou can appl Theorem. to obtain. Polnomial functions and rational functions are two of the three basic tpes of algebraic functions. The following theorem deals with the it of the third tpe of algebraic function one that involves a radical. See Appendi A for a proof of this theorem. THE SQUARE ROOT SYMBOL The first use of a smbol to denote the square root can be traced to the siteenth centur. Mathematicians first used the smbol, which had onl two strokes. This smbol was chosen because it resembled a lowercase r, to stand for the Latin word radi, meaning root. THEOREM. The Limit of a Function Involving a Radical Let n be a positive integer. The following it is valid for all c if n is odd, and is valid for c > 0 if n is even. n c n c

21 60_00.qd //0 : PM Page 6 SECTION. Evaluating Limits Analticall 6 The following theorem greatl epands our abilit to evaluate its because it shows how to analze the it of a composite function. See Appendi A for a proof of this theorem. THEOREM.5 The Limit of a Composite Function If f and g are functions such that g L and f fl, then c fg f c g c f L. L EXAMPLE The Limit of a Composite Function a. Because 0 and 0 it follows that. 0 b. Because and it follows that You have seen that the its of man algebraic functions can be evaluated b direct substitution. The si basic trigonometric functions also ehibit this desirable qualit, as shown in the net theorem (presented without proof). THEOREM.6 Limits of Trigonometric Functions Let c be a real number in the domain of the given trigonometric function.. sin sin c. c. tan tan c. c cos cos c c cot cot c c 5. sec sec c 6. csc csc c c c EXAMPLE 5 Limits of Trigonometric Functions a. tan tan0 0 0 b. cos cos cos c. 0 sin 0 sin 0 0

22 60_00.qd //0 : PM Page 6 6 CHAPTER Limits and Their Properties A Strateg for Finding Limits On the previous three pages, ou studied several tpes of functions whose its can be evaluated b direct substitution. This knowledge, together with the following theorem, can be used to develop a strateg for finding its. A proof of this theorem is given in Appendi A. f() = THEOREM.7 Functions That Agree at All But One Point Let c be a real number and let f g for all c in an open interval containing c. If the it of g as approaches c eists, then the it of f also eists and f g. c c EXAMPLE 6 Finding the Limit of a Function Find the it:. g() = + + f and g agree at all but one point. Figure.7 Solution Let f. B factoring and dividing out like factors, ou can rewrite f as f So, for all -values other than, the functions f and g agree, as shown in Figure.7. Because g eists, ou can appl Theorem.7 to conclude that f and g have the same it at. g, Factor.. Divide out like factors. Appl Theorem.7. Use direct substitution. Simplif. STUDY TIP When appling this strateg for finding a it, remember that some functions do not have a it (as approaches c). For instance, the following it does not eist. A Strateg for Finding Limits. Learn to recognize which its can be evaluated b direct substitution. (These its are listed in Theorems. through.6.). If the it of f as approaches c cannot be evaluated b direct substitution, tr to find a function g that agrees with f for all other than c. [Choose g such that the it of g can be evaluated b direct substitution.]. Appl Theorem.7 to conclude analticall that f g gc. c c. Use a graph or table to reinforce our conclusion.

23 60_00.qd //0 : PM Page 6 SECTION. Evaluating Limits Analticall 6 Dividing Out and Rationalizing Techniques Two techniques for finding its analticall are shown in Eamples 7 and 8. The first technique involves dividing out common factors, and the second technique involves rationalizing the numerator of a fractional epression. EXAMPLE 7 Dividing Out Technique Find the it: 6. (, 5) 5 f is undefined when. Figure.8 NOTE In the solution of Eample 7, be sure ou see the usefulness of the Factor Theorem of Algebra. This theorem states that if c is a zero of a polnomial function, c is a factor of the polnomial. So, if ou appl direct substitution to a rational function and obtain rc pc qc 0 0 ou can conclude that c must be a common factor to both p and q. f() = Solution Although ou are taking the it of a rational function, ou cannot appl Theorem. because the it of the denominator is 0. 6 Direct substitution fails. Because the it of the numerator is also 0, the numerator and denominator have a common factor of. So, for all, ou can divide out this factor to obtain f 6 Using Theorem.7, it follows that g, Appl Theorem Use direct substitution. This result is shown graphicall in Figure.8. Note that the graph of the function f coincides with the graph of the function g, ecept that the graph of f has a gap at the point, 5. In Eample 7, direct substitution produced the meaningless fractional form 00. An epression such as 00 is called an indeterminate form because ou cannot (from the form alone) determine the it. When ou tr to evaluate a it and encounter this form, remember that ou must rewrite the fraction so that the new denominator does not have 0 as its it. One wa to do this is to divide out like factors, as shown in Eample 7. A second wa is to rationalize the numerator, as shown in Eample 8. TECHNOLOGY PITFALL Because the graphs of δ 5 + ε + δ f 6 and g Incorrect graph of Figure.9 f Glitch near (, 5) 5 ε differ onl at the point, 5, a standard graphing utilit setting ma not distinguish clearl between these graphs. However, because of the piel configuration and rounding error of a graphing utilit, it ma be possible to find screen settings that distinguish between the graphs. Specificall, b repeatedl zooming in near the point, 5 on the graph of f, our graphing utilit ma show glitches or irregularities that do not eist on the actual graph. (See Figure.9.) B changing the screen settings on our graphing utilit ou ma obtain the correct graph of f.

24 60_00.qd //0 : PM Page 6 6 CHAPTER Limits and Their Properties EXAMPLE 8 Rationalizing Technique Find the it: 0. Solution B direct substitution, ou obtain the indeterminate form Direct substitution fails. 0 0 f() = + In this case, ou can rewrite the fraction b rationalizing the numerator., 0 Now, using Theorem.7, ou can evaluate the it as shown. 0 0 The it of f as approaches 0 is Figure.0. A table or a graph can reinforce our conclusion that the it is (See Figure.0.). approaches 0 from the left. approaches 0 from the right. f ? f approaches 0.5. f approaches 0.5. NOTE The rationalizing technique for evaluating its is based on multiplication b a convenient form of. In Eample 8, the convenient form is.

25 60_00.qd //0 : PM Page 65 SECTION. Evaluating Limits Analticall 65 f g h h() f() g() The Squeeze Theorem Figure. f lies in here. g c h f The Squeeze Theorem The net theorem concerns the it of a function that is squeezed between two other functions, each of which has the same it at a given -value, as shown in Figure.. (The proof of this theorem is given in Appendi A.) THEOREM.8 The Squeeze Theorem If h f g for all in an open interval containing c, ecept possibl at c itself, and if h L g c c then f eists and is equal to L. c You can see the usefulness of the Squeeze Theorem in the proof of Theorem.9. THEOREM.9 Two Special Trigonometric Limits sin cos (cos θ, sin θ) (, tan θ) Proof To avoid the confusion of two different uses of, the proof is presented using the variable, where is an acute positive angle measured in radians. Figure. shows a circular sector that is squeezed between two triangles. FOR FURTHER INFORMATION For more information on the function f sin, see the article The Function sin b William B. Gearhart and Harris S. Shultz in The College Mathematics Journal. To view this article, go to the website θ (, 0) A circular sector is used to prove Theorem.9. Figure. tan θ sin θ θ θ θ Area of triangle Area of sector Area of triangle tan sin Multipling each epression b sin produces cos sin and taking reciprocals and reversing the inequalities ields cos sin. Because cos cos and sin sin, ou can conclude that this inequalit is valid for all nonzero in the open interval,. Finall, because cos and, ou can appl the Squeeze Theorem to 0 0 conclude that sin. The proof of the second it is left as an eercise (see 0 Eercise 0).

26 60_00.qd //0 : PM Page CHAPTER Limits and Their Properties EXAMPLE 9 A Limit Involving a Trigonometric Function Find the it: tan 0. f() = tan The it of f as approaches 0 is. Figure. Solution Direct substitution ields the indeterminate form 00. To solve this problem, ou can write tan as sin cos and obtain tan 0 0 sin cos. Now, because sin 0 ou can obtain. (See Figure..) and 0 tan 0 sin 0 0 cos cos EXAMPLE 0 A Limit Involving a Trigonometric Function Find the it: 0 sin. g() = sin 6 The it of g as approaches 0 is. Figure. Solution Direct substitution ields the indeterminate form 00. To solve this problem, ou can rewrite the it as sin 0 sin 0. Multipl and divide b. Now, b letting and observing that 0 if and onl if 0, ou can write sin 0 sin 0. (See Figure..) sin 0 TECHNOLOGY Use a graphing utilit to confirm the its in the eamples and eercise set. For instance, Figures. and. show the graphs of f tan and g sin. Note that the first graph appears to contain the point 0, and the second graph appears to contain the point 0,, which lends support to the conclusions obtained in Eamples 9 and 0.

27 60_00.qd //0 : PM Page 67 SECTION. Evaluating Limits Analticall 67 Eercises for Section. In Eercises, use a graphing utilit to graph the function and visuall estimate the its.. h 5. g (a) (b) In Eercises 5, find the it h 5 (a) (b) h (b). f cos. (a) f (a) 0 f (b) 0 ft t 9 g g f t t t ft t tan 6. In Eercises 7 0, use the information to evaluate the its. 7. f 8. c c (a) c (b) c (c) (d) 9. f 0. c (a) (b) (c) (d) See for worked-out solutions to odd-numbered eercises. g c c c f c f c f c 5g f g f g f g f (c) (d) f 7 c (a) f In Eercises, use the graph to determine the it visuall (if it eists). Write a simpler function that agrees with the given function at all but one point... h g sec 7 6 f c g c (a) c (b) c (b) (c) (d) c c c c c f c f f g f g f g f 8 f In Eercises 6, find the its.. f 5, g (a) f (b) g (c). f 7, g (a) f (b) g (c) 5. f, g (a) f (b) g (c) 6. f, g 6 (a) f (b) g (c) In Eercises 7 6, find the it of the trigonometric function. 7. sin 8. tan sin. sec. cos 0. sin cos g f g f g f g f (a) (b) (a) (b) g 0 g g g (a) (b). g. f 5 h h 0 (a) f (b) f 0

28 60_00.qd //0 : PM Page CHAPTER Limits and Their Properties In Eercises 5 8, find the it of the function (if it eists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utilit to confirm our result In Eercises 9 6, find the it (if it eists) Graphical, Numerical, and Analtic Analsis In Eercises 6 66, use a graphing utilit to graph the function and estimate the it. Use a table to reinforce our conclusion. Then find the it b analtic methods In Eercises 67 78, determine the it of the trigonometric function (if it eists). 67. sin sin cos sin cos h h 0 h cos cot sin t t 0 t sin 0 sin sin Hint: Find 0 cos 0 cos tan 0 tan 0 sec tan sin cos sin. Graphical, Numerical, and Analtic Analsis In Eercises 79 8, use a graphing utilit to graph the function and estimate the it. Use a table to reinforce our conclusion. Then find the it b analtic methods. sin t cos t 0 t sin sin In Eercises 8 86, find 8. f 8. f 85. f 86. f In Eercises 87 and 88, use the Squeeze Theorem to find f. c 87. c 0 f 88. c a b a f b a In Eercises 89 9, use a graphing utilit to graph the given function and the equations and in the same viewing window. Using the graphs to observe the Squeeze Theorem visuall, find f. 89. f cos 90. f sin 9. f sin 9. f cos 9. f sin 9. h cos 99. Writing Use a graphing utilit to graph f, g sin, and h sin in the same viewing window. Compare the magnitudes of f and g when is close to 0. Use the comparison to write a short paragraph eplaining wh h Writing About Concepts f f. 95. In the contet of finding its, discuss what is meant b two functions that agree at all but one point. 96. Give an eample of two functions that agree at all but one point. 97. What is meant b an indeterminate form? 98. In our own words, eplain the Squeeze Theorem.

29 60_00.qd //0 : PM Page 69 SECTION. Evaluating Limits Analticall Writing Use a graphing utilit to graph f, g sin, and h sin in the same viewing window. Compare the magnitudes of f and g when is close to 0. Use the comparison to write a short paragraph eplaining wh h 0. 0 Free-Falling Object In Eercises 0 and 0, use the position function st 6t 000, which gives the height (in feet) of an object that has fallen for t seconds from a height of 000 feet. The velocit at time t a seconds is given b sa st. t a a t 0. If a construction worker drops a wrench from a height of 000 feet, how fast will the wrench be falling after 5 seconds? 0. If a construction worker drops a wrench from a height of 000 feet, when will the wrench hit the ground? At what velocit will the wrench impact the ground? Free-Falling Object In Eercises 0 and 0, use the position function st.9t 50, which gives the height (in meters) of an object that has fallen from a height of 50 meters. The velocit at time t a seconds is given b sa st. t a a t 0. Find the velocit of the object when t. 0. At what velocit will the object impact the ground? 05. Find two functions f and g such that f and g do 0 0 not eist, but f g does eist Prove that if f eists and f g does not c c eist, then g does not eist. c 07. Prove Propert of Theorem Prove Propert of Theorem.. (You ma use Propert of Theorem..) 09. Prove Propert of Theorem.. 0. Prove that if f 0, then f 0. c c. Prove that if f 0 and for a fied number c g M M and all c, then fg 0.. (a) Prove that if c f 0, then f 0. c c (Note: This is the converse of Eercise 0.) (b) Prove that if f L, then c Hint: Use the inequalit c f L. f L f L. True or False? In Eercises 8, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false.. 0 sin. 5. If f g for all real numbers other than 0, and f L, 0 then 6. If f L, then f c L. c 7. f, where 8. If f < g for all a, then f < g. a a 9. Think About It Find a function f to show that the converse of Eercise (b) is not true. [Hint: Find a function f such that f L but f does not eist.] c c 0. Prove the second part of Theorem.9 b proving that cos Let f 0,, and g 0,, Find (if possible) if is rational if is irrational if is rational if is irrational. 0 g L. 0 f, 0, f and > g. 0. Graphical Reasoning Consider f sec. (a) Find the domain of f. (b) Use a graphing utilit to graph f. Is the domain of f obvious from the graph? If not, eplain. (c) Use the graph of f to approimate f. 0 (d) Confirm the answer in part (c) analticall.. Approimation cos (a) Find. 0 (b) Use the result in part (a) to derive the approimation cos for near 0. (c) Use the result in part (b) to approimate cos0.. (d) Use a calculator to approimate cos0. to four decimal places. Compare the result with part (c).. Think About It When using a graphing utilit to generate a table to approimate sin, a student concluded that 0 the it was rather than. Determine the probable cause of the error.

30 60_00.qd //0 : PM Page CHAPTER Limits and Their Properties Section. EXPLORATION Informall, ou might sa that a function is continuous on an open interval if its graph can be drawn with a pencil without lifting the pencil from the paper. Use a graphing utilit to graph each function on the given interval. From the graphs, which functions would ou sa are continuous on the interval? Do ou think ou can trust the results ou obtained graphicall? Eplain our reasoning. Function Interval a., b., c. sin, d.,, 0 e.,, > 0 Continuit and One-Sided Limits Determine continuit at a point and continuit on an open interval. Determine one-sided its and continuit on a closed interval. Use properties of continuit. Understand and use the Intermediate Value Theorem. Continuit at a Point and on an Open Interval In mathematics, the term continuous has much the same meaning as it has in everda usage. Informall, to sa that a function f is continuous at c means that there is no interruption in the graph of f at c. That is, its graph is unbroken at c and there are no holes, jumps, or gaps. Figure.5 identifies three values of at which the graph of f is not continuous. At all other points in the interval a, b, the graph of f is uninterrupted and continuous. a f(c) is not defined. c Three conditions eist for which the graph of Figure.5 b a f() c does not eist. c In Figure.5, it appears that continuit at c can be destroed b an one of the following conditions.. The function is not defined at c.. The it of f does not eist at c.. The it of f eists at c, but it is not equal to fc. If none of the above three conditions is true, the function f is called continuous at c, as indicated in the following important definition. b f is not continuous at c. a f() f(c) c c b FOR FURTHER INFORMATION For more information on the concept of continuit, see the article Leibniz and the Spell of the Continuous b Hard Grant in The College Mathematics Journal. To view this article, go to the website Definition of Continuit Continuit at a Point: A function f is continuous at c if the following three conditions are met.. fc is defined.. f eists. c. f f c. c Continuit on an Open Interval: A function is continuous on an open interval a, b if it is continuous at each point in the interval. A function that is continuous on the entire real line, is everwhere continuous.