Limits and Their Properties

Transcription

1 Limits and Their Properties The it of a function is the primar concept that distinguishes calculus from algebra and analtic geometr. The notion of a it is fundamental to the stud of calculus. Thus, it is important to acquire a good working knowledge of its before moving on to other topics in calculus. In this chapter, ou should learn the following. How calculus compares with precalculus. (.) How to find its graphicall and numericall. (.) How to evaluate its analticall. (.) How to determine continuit at a point and on an open interval, and how to determine one-sided its. (.) How to determine infinite its and find vertical asmptotes. (.5) European Space Agenc/NASA According to NASA, the coldest place in the known universe is the Boomerang nebula. The nebula is five thousand light ears from Earth and has a temperature of 7C. That is onl warmer than absolute zero, the coldest possible temperature. How did scientists determine that absolute zero is the lower it of the temperature of matter? (See Section., Eample 5.) f() = + f() = f is undefined at = 0. + 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. (See Section..)

2 Chapter Limits and Their Properties. 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. 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? What Is Calculus? Calculus is the mathematics of change. For instance, calculus is the mathematics of velocities, accelerations, 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. The curvature of a circle is constant and can be analzed with precalculus mathematics. To analze the variable curvature of a general curve, 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 are listed some familiar precalculus concepts coupled with their calculus counterparts. 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 follows. Chapter P: Preparation for Calculus Precalculus Chapter : Limits and Their Properties Limit process Chapter : Differentiation Calculus

3 . A Preview of Calculus Without Calculus With Differential Calculus = f() Value of f Limit of f as when c approaches c c c = f() Slope of a line Δ Slope of a curve d Δ 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 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 . 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 f, the slope of the secant line through these two points can be found using precalculus and is given b m sec f c f c c c f c f c. Q(c + Δ, f(c + Δ)) P(c, f(c)) f(c + Δ) f(c) Q Secant lines Δ P Tangent line The Mistress Fellows, Girton College, Cambridge 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) The secant line through c, fc and (b) As Q approaches P, the secant lines c, fc approach the tangent line. Figure. As point Q approaches point P, the slopes of the secant lines approach 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 slopes of the secant lines. (Much more will be said about this important calculus concept in Chapter.) EXPLORATION The following points lie on the graph of f. Q.5, f.5, Q.00, f.00, Q., f., Q.0, f.0, Q 5.000, f.000 Each successive point gets closer to the point P,. Find the slopes of the secant lines 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 6 Chapter Limits and Their Properties The Area Problem a = f() b 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. Area under a curve Figure. = 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 Approimation using four rectangles Figure. b a Approimation using eight rectangles b 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 . A Preview of Calculus 7. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 5, 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 at. 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 at. 5. Find the area of the shaded region. (a) (b) 5 (, ) (0, 0) ( ) f() = (5, 0) 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. (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of f at P,. Describe how to improve our approimation of the slope. 7. Secant Lines Consider the function f 6 and the point P, 8 on the graph of f. (a) Graph f and the secant lines passing through P, 8 and Q, f for -values of,.5, and.5. (b) Find the slope of each secant line. (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of f at P, 8. Describe how to improve our approimation of the slope. f() = (a) Use the rectangles in each graph to approimate the area of the region bounded b sin, 0, 0, and. (b) Describe how ou could continue this process to obtain a more accurate approimation of the area. 9. (a) Use the rectangles in each graph to approimate the area of the region bounded b 5, 0,, and 5. 5 π π 5 (b) Describe how ou could continue this process to obtain a more accurate approimation of the area. CAPSTONE 0. How would ou describe the instantaneous rate of change of an automobile s position on the highwa? 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 π (, 5) π 5 5 (5, )

8 8 Chapter Limits and Their Properties. 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 definition of 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 . 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. approaches 0 from the left. approaches 0 from the right. f() = f ? The it of f as approaches 0 is. Figure.6 f approaches. f approaches. 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., f() = 0, = The it of f as approaches is. Figure.7 EXAMPLE Finding a Limit Find the it of f as approaches, where f is defined as f, 0, 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 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. δ δ f() = f does not eist. 0 Figure.8 f() = f() = EXAMPLE Behavior That Differs from the Right and from the Left Show that the it does not eist. Solution Consider the graph of the function From Figure.8 and the definition of absolute value,, ou can see that,, Definition of absolute value This means that no matter how close gets to 0, there will be both positive and negative -values that ield f or 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 follows., 0 0 if 0 if < 0 if > 0 if < 0. 0, f. Negative -values Positive -values.. ield ield Because approaches a different number from the right side of 0 than it approaches from the left side, the it does not eist. 0 f() = EXAMPLE Unbounded Behavior Discuss the eistence of the it 0. 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 . Finding Limits Graphicall and Numericall 5 EXAMPLE 5 Oscillating Behavior f() = sin Discuss the eistence of the it sin 0. 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 sin / Limit does not eist. COMMON TYPES OF BEHAVIOR ASSOCIATED WITH NONEXISTENCE OF A LIMIT f does not eist. 0 Figure.0. 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 This is 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 is attributed to the German mathematician Peter Gustav Dirichlet. Incorrect graph of Figure.. f sin. The icon indicates that ou will find a CAS Investigation on the book s website. The CAS Investigation is a collaborative eploration of this eample using the computer algebra sstems Maple and Mathematica.

12 5 Chapter Limits and Their Properties L + ε L L ε c + δ c c δ The - definition of the it of f as approaches c Figure. (c, L) A Formal Definition of Limit Let s take another look at the informal definition of 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 definition 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 79).

13 . 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 0 < <. 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 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 and thus.,, < 5 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 See for worked-out solutions to odd-numbered eercises. 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 f f f f f sin f cos f f

15 . Finding Limits Graphicall and Numericall 55 In Eercises 9, create a table of values for the function and use the result to estimate the it. Use a graphing utilit to graph the function to confirm our result.. cos. 0 tan sin tan. 0 0 tan π π π π In Eercises 5, use the graph to find the it (if it eists). If the it does not eist, eplain wh f f, 0, f 6 f,, In Eercises 5 and 6, 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. 5. (a) f (b) f (c) f (d) f 6. (a) f (b) f (c) f 0 (d) f 0 (e) f (f ) f (g) f (h) f In Eercises 7 and 8, use the graph of f to identif the values of c for which f eists c sin. sec 0 π π In Eercises 9 and 0, sketch the graph of f. Then identif the values of c for which f eists. 9. f, 8,, sin, 0. f cos, cos, c < < < 0 0 >

16 56 Chapter Limits and Their Properties In Eercises and, sketch a graph of a function f that satisfies the given values. (There are man correct answers.). f 0 is undefined.. f 0 f 0 f 6 f f does not eist.. Modeling Data For a long distance phone call, a hotel charges $9.99 for the first minute and $0.79 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 6. (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 f 0 f 0 t 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 f f then then C? (c) Use the graph to complete the table and observe the behavior of the function as t approaches. t C? Does the it of Ct as t approaches eist? Eplain.. Repeat Eercise for Ct t. 5. The graph of f is shown in the figure. Find such that if 0 < < then f < The graph of f is shown in the figure. Find such that if 0 < < then In Eercises 9, find the it L. Then find > 0 such that whenever 0 < c <. f L < f =. = = f < 0.. 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 . Finding Limits Graphicall and Numericall 57 In Eercises 5, find the it L. Then use the - definition to prove that the it is L What is the it of f as approaches? 56. What is the it of g as approaches? Writing In Eercises 57 60, 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 58. f f ) f 9 f 9 f 9 f 5 WRITING ABOUT CONCEPTS f 6. Write a brief description of the meaning of the notation f 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? 6. Identif three tpes of behavior associated with the noneistence of a it. Illustrate each tpe with a graph of a function. CAPSTONE 6. (a) If f, can ou conclude anthing about the it of f as approaches? Eplain our reasoning. (b) If the it of f as approaches is, can ou conclude anthing about f? Eplain our reasoning. 65. 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. 66. 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. 67. Consider the function f. Estimate the it 0 b evaluating f at -values near 0. Sketch the graph of f. 68. Consider the function f. Estimate 0 b evaluating f at -values near 0. Sketch the graph of f. 69. 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 58 Chapter Limits and Their Properties 70. 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. True or False? In Eercises 7 7, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 7. If f is undefined at c, then the it of f as approaches c does not eist. 7. 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 7. If f L, then f c L. c In Eercises 75 and 76, consider the function f. 75. Is 0.5 a true statement? Eplain Is 0 a true statement? Eplain. 0 sin n 77. Use a graphing utilit to evaluate the it for several 0 values of n. What do ou notice? tan n 78. Use a graphing utilit to evaluate the it for several 0 values of n. What do ou notice? 79. 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. 80. Consider the line f m b, where m 0. Use the definition of it to prove that f mc b. c 8. Prove that f L is equivalent to f L 0. c c - 8. (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. 8. 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. 8. Programming Use the program ou created in Eercise 8 to approimate the it PUTNAM EXAM CHALLENGE 85. 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 86. 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 . Evaluating Limits Analticall 59. 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: c f n L n

20 60 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 POLYNOMIAL 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. 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. 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. 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 . 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 and 0 0 it follows that. 0 b. Because it follows that 0 8. and 8 8 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 5. sec sec c 6. csc csc c c cos cos c c cot cot c c c EXAMPLE 5 Limits of Trigonometric Functions a. tan tan0 0 0 b. cos cos cos c. 0 sin sin 0 0 0

22 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 g() = + + f and g agree at all but one point. Figure.7 Find the it:. 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 STRATEGY 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 . Evaluating Limits Analticall 6 Dividing Out and Rationalizing Techniques Two techniques for finding its analticall are shown in Eamples 7 and 8. The dividing out technique involves dividing out common factors, and the rationalizing technique involves rationalizing the numerator of a fractional epression. EXAMPLE 7 Dividing Out Technique (, 5) 5 f is undefined when. Figure.8 f() = 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 of both p and q. 5 + ε δ + δ Incorrect graph of Figure.9 f Glitch near (, 5) 5 ε Find the it: 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 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 This is Because the graphs of f 6 and g 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 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 . 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 (also called the Sandwich Theorem or the Pinching 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. θ (, 0) A circular sector is used to prove Theorem.9. Figure. 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 tan θ. θ 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 ). θ sin θ

26 66 Chapter Limits and Their Properties EXAMPLE 9 A Limit Involving a Trigonometric Function f() = tan The it of f as approaches 0 is. Figure. tan Find the it: 0. 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 0 tan. (See Figure..) and 0 sin 0 0 cos cos EXAMPLE 0 A Limit Involving a Trigonometric Function g() = sin 6 The it of g as approaches 0 is. Figure. Find the it: 0 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. sin 0 Appl Theorem.9(). TECHNOLOGY Use a graphing utilit to confirm the its in the eamples and in the 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 . Evaluating Limits Analticall 67. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, use a graphing utilit to graph the function and visuall estimate the its.. h. g (a) (b) h (a) (b). f cos. (a) f (a) 0 (b) f (b) h In Eercises 5, find the it In Eercises 6, find the its.. f 5, g (a) (b) (c) g f. f 7, g (a) (b) (c) g f 5. f, g (a) (b) (c) g f 6. f, g 6 (a) (b) (c) g f In Eercises 7 6, find the it of the trigonometric function. 7. sin 8. tan sin. sec. cos 0. sin. cos tan 6. sec ft t 9 g g f t t t ft t 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) (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 (a) (b) (a) (b) g c c c f c f c f c g 0 g 5g f g f g f g f g g f c g c (a) c (b) c (b) (c) (d) (a) (b). g. f c c c c c f c h h 0 f f g f 8 f (a) f (b) f 0 f g f g

28 68 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) In Eercises 65 76, determine the it of the trigonometric function (if it eists). 65. sin sin cos 68. sin tan cos h h 0 h cos tan cot sin cos sin t t 0 t sin 0 sin sin Hint: Find 0 cos 0 cos tan 0 sec Graphical, Numerical, and Analtic Analsis In Eercises 77 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 sin t t 0 t sin 0 8. In Eercises 85 88, find 85. f 86. f 87. f 88. f In Eercises 89 and 90, use the Squeeze Theorem to find f. c 89. c 0 f 90. c a b a f b a In Eercises 9 96, 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. 9. f cos f sin 95. f sin 96. h cos 0. 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 5 cos 0 0 sin f f. f sin f cos 97. In the contet of finding its, discuss what is meant b two functions that agree at all but one point. 98. Give an eample of two functions that agree at all but one point. 99. What is meant b an indeterminate form? 00. In our own words, eplain the Squeeze Theorem.

29 . Evaluating Limits Analticall Writing Use a graphing utilit to graph 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. Free-Falling Object In Eercises 0 and 0, use the position function st 6t 500, which gives the height (in feet) of an object that has fallen for t seconds from a height of 500 feet. The velocit at time t a seconds is given b t a 0. If a construction worker drops a wrench from a height of 500 feet, how fast will the wrench be falling after seconds? 0. If a construction worker drops a wrench from a height of 500 feet, when will the wrench hit the ground? At what velocit will the wrench impact the ground? Free-Falling Object In Eercises 05 and 06, use the position function st.9t 00, which gives the height (in meters) of an object that has fallen from a height of 00 meters. The velocit at time t a seconds is given b t a f, g sin, and h sin sa st. a t sa st. a t 05. Find the velocit of the object when t. 06. At what velocit will the object impact the ground? 07. 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 09. Prove Propert of Theorem.. 0. Prove Propert of Theorem.. (You ma use Propert of Theorem..). Prove Propert of Theorem... Prove that if f 0, then f 0. c c. Prove that if f 0 and for a fied number c 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.) (b) Prove that if f L, then c Hint: Use the inequalit 5. 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 but f does not eist.] c f L c 0 g M c f L. f L f L. CAPSTONE 6. Let f, Find f. 5,. True or False? In Eercises 7, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. sin If f g for all real numbers other than 0, and f L, then g L If f L, then f c L.. f, where c. If f < g for all a, then f < g. a a. Prove the second part of Theorem.9. cos 0 0. Let f 0,, and g 0,, Find (if possible) f and 5. Graphical Reasoning Consider f (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 our answer to part (c) analticall. 6. Approimation (a) Find 0 if is rational if is irrational. 0 cos. 0 f, 0, if is rational if is irrational (b) Use our answer to part (a) to derive the approimation cos for near 0. (c) Use our answer to part (b) to approimate cos0.. (d) Use a calculator to approimate cos0. to four decimal places. Compare the result with part (c). 7. Think About It When using a graphing utilit to generate a table to approimate sin, a student concluded that 0 > g. 0 sec. the it was rather than. Determine the probable cause of the error.

30 70 Chapter Limits and Their Properties. 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. 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. a. b. c. d. Function sin Interval,,,,, 0 e.,, > 0 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 b Three conditions eist for which the graph of f is not continuous at c. Figure.5 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 three conditions above is true, the function f is called continuous at c, as indicated in the following important definition. b 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 CONTINUITY Continuit at a Point: A function f is continuous at c if the following three conditions are met.. fc is defined.. 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.

31 . Continuit and One-Sided Limits 7 a c b (a) Removable discontinuit a c b (b) Nonremovable discontinuit Consider an open interval I that contains a real number c. If a function f is defined on I (ecept possibl at c), and f is not continuous at c, then f is said to have a discontinuit at c. Discontinuities fall into two categories: removable and nonremovable. A discontinuit at c is called removable if f can be made continuous b appropriatel defining (or redefining) fc. For instance, the functions shown in Figures.6(a) and (c) have removable discontinuities at c and the function shown in Figure.6(b) has a nonremovable discontinuit at c. EXAMPLE Continuit of a Function Discuss the continuit of each function., a. b. c. h 0 g f d. sin, > 0 Solution a. The domain of f is all nonzero real numbers. From Theorem., ou can conclude that f is continuous at ever -value in its domain. At 0, f has a nonremovable discontinuit, as shown in Figure.7(a). In other words, there is no wa to define f0 so as to make the function continuous at 0. b. The domain of g is all real numbers ecept. From Theorem., ou can conclude that g is continuous at ever -value in its domain. At, the function has a removable discontinuit, as shown in Figure.7(b). If g is defined as, the newl defined function is continuous for all real numbers. c. The domain of h is all real numbers. The function h is continuous on, 0 and 0,, and, because h, h is continuous on the entire real line, as shown 0 in Figure.7(c). d. The domain of is all real numbers. From Theorem.6, ou can conclude that the function is continuous on its entire domain,,, as shown in Figure.7(d). a (c) Removable discontinuit Figure.6 c b f() = (, ) g() = (a) Nonremovable discontinuit at 0 (b) Removable discontinuit at = sin STUDY TIP Some people ma refer to the function in Eample (a) as discontinuous. We have found that this terminolog can be confusing. Rather than saing that the function is discontinuous, we prefer to sa that it has a discontinuit at 0. +, 0 h() = +, > 0 (c) Continuous on entire real line Figure.7 π π (d) Continuous on entire real line

32 7 Chapter Limits and Their Properties approaches c from the right. One-Sided Limits and Continuit on a Closed Interval To understand continuit on a closed interval, ou first need to look at a different tpe of it called a one-sided it. For eample, the it from the right (or right-hand it) means that approaches c from values greater than c [see Figure.8(a)]. This it is denoted as c < f L. c Limit from the right (a) Limit from right Similarl, the it from the left (or left-hand it) means that approaches c from values less than c [see Figure.8(b)]. This it is denoted as approaches c from the left. f L. c Limit from the left c > (b) Limit from left Figure.8 One-sided its are useful in taking its of functions involving radicals. For instance, if n is an even integer, n 0. 0 EXAMPLE A One-Sided Limit f() = Find the it of f as approaches from the right. Solution As shown in Figure.9, the it as approaches from the right is 0. One-sided its can be used to investigate the behavior of step functions. One common tpe of step function is the greatest integer function, defined b The it of f as approaches from the right is 0. Figure.9 greatest integer n such that n. Greatest integer function For instance,.5 and.5. EXAMPLE The Greatest Integer Function f() = [[ ]] Find the it of the greatest integer function f as approaches 0 from the left and from the right. Solution b As shown in Figure.0, the it as approaches 0 from the left is given 0 and the it as approaches 0 from the right is given b Greatest integer function Figure The greatest integer function has a discontinuit at zero because the left and right its at zero are different. B similar reasoning, ou can see that the greatest integer function has a discontinuit at an integer n.

33 . Continuit and One-Sided Limits 7 When the it from the left is not equal to the it from the right, the (twosided) it does not eist. The net theorem makes this more eplicit. The proof of this theorem follows directl from the definition of a one-sided it. THEOREM.0 THE EXISTENCE OF A LIMIT Let f be a function and let c and L be real numbers. The it of f as approaches c is L if and onl if f L and f L. c c The concept of a one-sided it allows ou to etend the definition of continuit to closed intervals. Basicall, a function is continuous on a closed interval if it is continuous in the interior of the interval and ehibits one-sided continuit at the endpoints. This is stated formall as follows. a Continuous function on a closed interval Figure. b DEFINITION OF CONTINUITY ON A CLOSED INTERVAL A function f is continuous on the closed interval [a, b] if it is continuous on the open interval a, b and f fa and f fb. a b The function f is continuous from the right at a and continuous from the left at b (see Figure.). Similar definitions can be made to cover continuit on intervals of the form a, b and a, b that are neither open nor closed, or on infinite intervals. For eample, the function f is continuous on the infinite interval 0,, and the function g is continuous on the infinite interval,. f() = f is continuous on,. Figure. EXAMPLE Continuit on a Closed Interval Discuss the continuit of f. Solution The domain of f is the closed interval,. At all points in the open interval,, the continuit of f follows from Theorems. and.5. Moreover, because and 0 f 0 f Continuous from the right Continuous from the left ou can conclude that f is continuous on the closed interval,, as shown in Figure..

34 7 Chapter Limits and Their Properties The net eample shows how a one-sided it can be used to determine the value of absolute zero on the Kelvin scale. EXAMPLE 5 Charles s Law and Absolute Zero Photo courtes of W. Ketterle, MIT V = 0.08T +. ( 7.5, 0) T The volume of hdrogen gas depends on its temperature. Figure. In 00, researchers at the Massachusetts Institute of Technolog used lasers and evaporation to produce a supercold gas in which atoms overlap. This gas is called a Bose-Einstein condensate. The measured a temperature of about 50 pk (picokelvin), or approimatel C. (Source: Science magazine, September, 00) V On the Kelvin scale, absolute zero is the temperature 0 K. Although temperatures ver close to 0 K have been produced in laboratories, absolute zero has never been attained. In fact, evidence suggests that absolute zero cannot be attained. How did scientists determine that 0 K is the lower it of the temperature of matter? What is absolute zero on the Celsius scale? Solution The determination of absolute zero stems from the work of the French phsicist Jacques Charles (76 8). Charles discovered that the volume of gas at a constant pressure increases linearl with the temperature of the gas. The table illustrates this relationship between volume and temperature. To generate the values in the table, one mole of hdrogen is held at a constant pressure of one atmosphere. The volume V is approimated and is measured in liters, and the temperature T is measured in degrees Celsius. T V The points represented b the table are shown in Figure.. Moreover, b using the points in the table, ou can determine that T and V are related b the linear equation T V NOTE V 0.08T. V RT or B reasoning that the volume of the gas can approach 0 (but can never equal or go below 0), ou can determine that the least possible temperature is given b V. T V 0 V Use direct substitution. So, absolute zero on the Kelvin scale 0 K is approimatel 7.5 on the Celsius scale. The following table shows the temperatures in Eample 5 converted to the Fahrenheit scale. Tr repeating the solution shown in Eample 5 using these temperatures and volumes. Use the result to find the value of absolute zero on the Fahrenheit scale. T Charles s Law for gases (assuming constant pressure) can be stated as Charles s Law V where V is volume, R is a constant, and T is temperature. In the statement of this law, what propert must the temperature scale have?

35 . Continuit and One-Sided Limits 75 Properties of Continuit In Section., ou studied several properties of its. Each of those properties ields a corresponding propert pertaining to the continuit of a function. For instance, Theorem. follows directl from Theorem.. (A proof of Theorem. is given in Appendi A.) Bettmann/Corbis AUGUSTIN-LOUIS CAUCHY ( ) The concept of a continuous function was first introduced b Augustin-Louis Cauch in 8. The definition given in his tet Cours d Analse stated that indefinite small changes in were the result of indefinite small changes in. f will be called a continuous function if the numerical values of the difference f f decrease indefinitel with those of. THEOREM. PROPERTIES OF CONTINUITY If b is a real number and f and g are continuous at c, then the following functions are also continuous at c.. Scalar multiple: bf. Sum or difference: f ± g. Product: fg f. Quotient: if gc 0 g, The following tpes of functions are continuous at ever point in their domains.. Polnomial: p a n n a n n... a a 0. Rational:. Radical: r p q, f n q 0. Trigonometric: sin, cos, tan, cot, sec, csc B combining Theorem. with this summar, ou can conclude that a wide variet of elementar functions are continuous at ever point in their domains. EXAMPLE 6 Appling Properties of Continuit B Theorem., it follows that each of the functions below is continuous at ever point in its domain. f sin, f tan, f cos The net theorem, which is a consequence of Theorem.5, allows ou to determine the continuit of composite functions such as f sin, f, f tan. NOTE One consequence of Theorem. is that if f and g satisf the given conditions, ou can determine the it of fg as approaches c to be fg fgc. c THEOREM. CONTINUITY OF A COMPOSITE FUNCTION If g is continuous at c and f is continuous at gc, then the composite function given b f g fg is continuous at c. PROOF B the definition of continuit, g gc and f f gc. c gc Appl Theorem.5 with L gc to obtain fg f So, c g c fgc. f g fg is continuous at c.