1.5 LIMITS. The Limit of a Function

Transcription

1 60040_005.qd /5/05 :0 PM Page 49 SECTION.5 Limits 49.5 LIMITS Find its of functions graphicall and numericall. Use the properties of its to evaluate its of functions. Use different analtic techniques to evaluate its of functions. Evaluate one-sided its. Recognize unbounded behavior of functions. The Limit of a Function In everda language, people refer to a speed it, a wrestler s weight it, the it of one s endurance, or stretching a spring to its it. These phrases all suggest that a it is a bound, which on some occasions ma not be reached but on other occasions ma be reached or eceeded. Consider a spring that will break onl if a weight of 0 pounds or more is attached. To determine how far the spring will stretch without breaking, ou could attach increasingl heavier weights and measure the spring length s for each weight w, as shown in Figure.5. If the spring length approaches a value of L, then it is said that the it of s as w approaches 0 is L. A mathematical it is much like the it of a spring. The notation for a it is f L c which is read as the it of f as approaches c is L. EXAMPLE Find the it:. Finding a Limit SOLUTION Let f. From the graph of f in Figure.5, it appears that f approaches as approaches from either side, and ou can write. The table ields the same conclusion. Notice that as gets closer and closer to, f gets closer and closer to. approaches. approaches. w = 0 w = w = 7.5 w = 9.5 w = FIGURE.5 What is the it of s as w approaches 0 pounds 4 s ( + ) = (, ) f f approaches. f approaches. FIGURE.5 TRY IT Find the it: 4.

2 60040_005.qd /5/05 :0 PM Page CHAPTER Functions, Graphs, and Limits EXAMPLE Finding Limits Graphicall and Numericall (, ) = Find the it: (a) f (b) (c) SOLUTION f. f f, 0, (a) (a) From the graph of f, in Figure.5(a), it appears that f approaches as approaches from either side. A missing point is denoted b the open dot on the graph. This conclusion is reinforced b the table. Be sure ou see that it does not matter that f is undefined when. The it depends onl on values of f near, not at. does not eist. (, ) approaches. approaches f f approaches. f approaches. (b) (, ) (b) From the graph of f, in Figure.5(b), ou can see that f for all values to the left of and f for all values to the right of. So, f is approaching a different value from the left of than it is from the right of. In such situations, we sa that the it does not eist. This conclusion is reinforced b the table. f() = approaches. approaches (, ) f (c) FIGURE.5 f approaches. f approaches. (c) From the graph of f, in Figure.5(c), it appears that f approaches as approaches from either side. This conclusion is reinforced b the table. It does not matter that f 0. The it depends onl on values of f near, not at. TRY IT approaches. approaches. Find the it: (a) (b) f 4 f (c) f, 0, f f f approaches. f approaches.

3 60040_005.qd /5/05 :0 PM Page 5 SECTION.5 Limits 5 There are three important ideas to learn from Eamples and.. Saing that the it of f approaches L as approaches c means that the value of f ma be made arbitraril close to the number L b choosing closer and closer to c.. For a it to eist, ou must allow to approach c from either side of c. If f approaches a different number as approaches c from the left than it does as approaches c from the right, then the it does not eist. [See Eample (b).]. The value of f when c has no bearing on the eistence or noneistence of the it of f as approaches c. For instance, in Eample (a), the it of f eists as approaches even though the function f is not defined at. Definition of the Limit of a Function If f becomes arbitraril close to a single number L as approaches c from either side, then f L c which is read as the it of f as approaches c is L. TECHNOLOGY Tr using a graphing utilit to determine the following it. 4 5 You can do this b graphing f 4 5 and zooming in near. From the graph, what does the it appear to be Properties of Limits Man times the it of f as approaches c is simpl f c, as shown in Eample. Whenever the it of f as approaches c is f f c c Substitute c for. the it can be evaluated b direct substitution. (In the net section, ou will learn that a function that has this propert is continuous at c.) It is important that ou learn to recognize the tpes of functions that have this propert. Some basic ones are given in the following list. Properties of Limits Let b and c be real numbers, and let n be a positive integer.. b b c. c c. c n c n 4. n c n c In Propert 4, if n is even, then c must be positive.

4 60040_005.qd /5/05 :0 PM Page 5 5 CHAPTER Functions, Graphs, and Limits B combining the properties of its with the rules for operating with its shown below, ou can find its for a wide variet of algebraic functions. TECHNOLOGY Smbolic computer algebra sstems are capable of evaluating its. Tr using a computer algebra sstem to evaluate the it given in Eample. Operations with 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 and g K c c. Scalar multiple:. Sum or difference:. Product: f g LK c f 4. Quotient: provided K 0 c g L K, 5. Power: c f n L n 6. Radical: n f L n c bf bl c f ± g L ± K c In Propert 6, if n is even, then L must be positive. DISCOVERY Use a graphing utilit to graph. Does approach a it as approaches 0 Evaluate at several positive and negative values of near 0 to confirm our answer. Does eist EXAMPLE Find the it: Finding the Limit of a Polnomial Function Appl Propert. Use direct substitution. Simplif. TRY IT Find the it: 4. Eample is an illustration of the following important result, which states that the it of a polnomial can be evaluated b direct substitution. The Limit of a Polnomial Function If p is a polnomial function and c is an real number, then p p c. c

5 60040_005.qd /5/05 :0 PM Page 5 SECTION.5 Limits 5 Techniques for Evaluating Limits Man techniques for evaluating its are based on the following important theorem. Basicall, the theorem states that if two functions agree at all but a single point c, then the have identical it behavior at c. The Replacement Theorem Let c be a real number and let f g for all c. If the it of g eists as c, then the it of f also eists and f g. c c To appl the Replacement Theorem, ou can use a result from algebra which states that for a polnomial function p, p c 0 if and onl if c is a factor of p. This concept is demonstrated in Eample 4. EXAMPLE 4 Find the it:. Finding the Limit of a Function SOLUTION Note that the numerator and denominator are zero when. This implies that is a factor of both, and ou can divide out this like factor. Factor numerator. Divide out like factor., Simplif. So, the rational function and the polnomial function agree for all values of other than, and ou can appl the Replacement Theorem. Figure.54 illustrates this result graphicall. Note that the two graphs are identical ecept that the graph of g contains the point,, whereas this point is missing on the graph of f. (In the graph of f in Figure.54, the missing point is denoted b an open dot.) g() = + + FIGURE.54 f() = TRY IT 4 Find the it: 8. The technique used to evaluate the it in Eample 4 is called the dividing out technique. This technique is further demonstrated in the net eample.

6 60040_005.qd /5/05 :0 PM Page CHAPTER Functions, Graphs, and Limits DISCOVERY Use a graphing utilit to graph 6. Is the graph a line Wh or wh not (, 5) 4 5 FIGURE.55. f() = f is undefined when EXAMPLE 5 Find the it: Using the Dividing Out Technique SOLUTION Direct substitution fails because both the numerator and the denominator are zero when However, because the its of both the numerator and denominator are zero, ou know that the have a common factor of. So, for all, ou can divide out this factor to obtain the following Factor numerator. Divide out like factor. Simplif. 5 Direct substitution This result is shown graphicall in Figure.55. Note that the graph of f coincides with the graph of g, ecept that the graph of f has a hole at, 5. TRY IT 5 Find the it:. STUDY TIP When ou tr to evaluate a it and both the numerator and denominator are zero, 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 5. Another technique is to rationalize the numerator, as shown in Eample 6. TRY IT 6 4 Find the it:. 0 EXAMPLE 6 Find the it: 0 Finding a Limit of a Function. SOLUTION Direct substitution fails because both the numerator and the denominator are zero when 0. In this case, ou can rewrite the fraction b rationalizing the numerator., 0 Now, using the Replacement Theorem, ou can evaluate the it as shown. 0 0

7 60040_005.qd /5/05 :0 PM Page 55 SECTION.5 Limits 55 One-Sided Limits In Eample (b), ou saw that one wa in which a it can fail to eist is when a function approaches a different value from the left of c than it approaches from the right of c. This tpe of behavior can be described more concisel with the concept of a one-sided it. f L c Limit from the left f L c Limit from the right The first of these two its is read as the it of f as approaches c from the left is L. The second is read as the it of f as approaches c from the right is L. EXAMPLE 7 Finding One-Sided Limits Find the it as 0 from the left and the it as 0 from the right for the function f. SOLUTION From the graph of f, shown in Figure.56, ou can see that f for all < 0. So, the it from the left is. Limit from the left 0 f() = Because f for all > 0, the it from the right is. Limit from the right 0 FIGURE.56 TRY IT 7 Find each it. (a) (b) In Eample 7, note that the function approaches different its from the left and from the right. In such cases, the it of f as c does not eist. For the it of a function to eist as c, both one-sided its must eist and must be equal. TECHNOLOGY On most graphing utilities, the absolute value function is denoted b abs. You can verif the result in Eample 7 b graphing abs in the viewing window and. Eistence of a Limit If f is a function and c and L are real numbers, then f L c if and onl if both the left and right its are equal to L.

8 60040_005.qd /5/05 :0 PM Page CHAPTER Functions, Graphs, and Limits 4 f() = 4 ( < ) f() = FIGURE.57 TRY IT 8 f() = 4 ( > ) Find the it of f as approaches 0. f, < 0, > EXAMPLE 8 Finding One-Sided Limits Find the it of f as approaches. f 4, 4, SOLUTION Remember that ou are concerned about the value of f near rather than at. So, for <, f is given b 4, and ou can use direct substitution to obtain f 4 For >, f is given b 4, and ou can use direct substitution to obtain f 4 Because both one-sided its eist and are equal to, it follows that f. < > The graph in Figure.57 confirms this conclusion. Shipping costs (in dollars) Deliver Service Rates For <, f() = For <, f() = 0 For 0 <, f() = 8 FIGURE.58 Weight (in pounds) EXAMPLE 9 Comparing One-Sided Limits An overnight deliver service charges $8 for the first pound and $ for each additional pound. Let represent the weight of a parcel and let f represent the shipping cost. f 8, 0,, Show that the it of f as does not eist. SOLUTION The graph of f is shown in Figure.58. The it of f as approaches from the left is f 0 whereas the it of f as approaches from the right is f. 0 < < < Because these one-sided its are not equal, the it of f as does not eist. TRY IT 9 Show that the it of f as does not eist in Eample 9.

9 60040_005.qd /5/05 :0 PM Page 57 SECTION.5 Limits 57 Unbounded Behavior Eample 9 shows a it that fails to eist because the its from the left and right differ. Another important wa in which a it can fail to eist is when f increases or decreases without bound as approaches c. EXAMPLE 0 An Unbounded Function Find the it (if possible). SOLUTION From Figure.59, ou can see that f decreases without bound as approaches from the left and f increases without bound as approaches from the right. Smbolicall, ou can write this as and. f() as f() as f() = Because f is unbounded as approaches, the it does not eist. FIGURE.59 TRY IT 0 Find the it (if possible): 5. STUDY TIP The equal sign in the statement f does not mean that the it c eists. On the contrar, it tells ou how the it fails to eist b denoting the unbounded behavior of f as approaches c. T AKE ANOTHER LOOK Evaluating a Limit Consider the it from Eample 6. 0 a. Approimate the it graphicall, using a graphing utilit. b. Approimate the it numericall, b constructing a table. c. Find the it analticall, using the Replacement Theorem, as shown in Eample 6. Of the three methods, which do ou prefer Eplain our reasoning.

10 60040_005.qd /5/05 :0 PM Page CHAPTER Functions, Graphs, and Limits PREREQUISITE REVIEW.5 The following warm-up eercises involve skills that were covered in earlier sections. You will use these skills in the eercise set for this section. In Eercises 4, evaluate the epression and simplif.. f (a) f (b) f c (c) f h.. 4. f,, f f 4 < (a) f (b) f (c) f t f h f h f h f h In Eercises 5 8, find the domain and range of the function and sketch its graph. 5. h 5 6. f g 5 f In Eercises 9 and 0, determine whether is a function of EXERCISES.5 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 f

11 60040_005.qd /5/05 :0 PM Page 59 SECTION.5 Limits In Eercises 9, use the graph to find the it (if it eists) (, ) (a) 0 (b).. (a) (, ) (b) In Eercises and 4, find the it of (a) f g, (b) f g, and (c) f g as approaches c.. f 4. c In Eercises 5 and 6, find the it of (a) f, (b) f, and (c) f as approaches c. 5. f 6 6. c f f = f() (0, ) = g() g 0 g g 9 c (0, ) (a) (b) (a) (b) f f (0, ) (, 5) h h 0 c f c g f 9 c = f() (, ) = h() (, 0) In Eercises 7, use the graph to find the it (if it eists). (a) (b) (c) f c f c c 7. = f() 8. = f() c =. = f(). In Eercises 40, find the it = f() (, ) (, ) (, ) (, ) c = (, 0) c = (, ) (, ) (, ) (, ) (, 0) c = = f() c = c = = f()

12 60040_005.qd /5/05 :0 PM Page CHAPTER Functions, Graphs, and Limits In Eercises 4 58, find the it (if it eists) t 5 t 46. t t 5 t 5 t t t t t t t 5t t 0 t t t t t t 4t t 0 t Graphical, Numerical, and Analtic Analsis In Eercises 59 6, 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 6 66, use a graphing utilit to estimate the it (if it eists) f, where f, 5, f s, where f s s s, s, s s > > The it of f is a natural base for man business applications, as ou will see in Section e.78 (a) Show the reasonableness of this it b completing the table f (b) Use a graphing utilit to graph f and to confirm the answer in part (a). (c) Find the domain and range of the function. 68. Find f, given 0 4 f 4, for all. 69. Environment The cost (in dollars) of removing p% of the pollutants from the water in a small lake is given b C 5,000p 00 p, where C is the cost and p is the percent of pollutants. (a) Find the cost of removing 50% of the pollutants. (b) What percent of the pollutants can be removed for $00,000 (c) Evaluate C. Eplain our results. p Compound Interest You deposit $000 in an account that is compounded quarterl at an annual rate of r (in decimal form). The balance A after 0 ears is A 000 r Does the it of A eist as the interest rate approaches 6% If so, what is the it 7. Compound Interest Consider a certificate of deposit that pas 0% (annual percentage rate) on an initial deposit of $500. The balance A after 0 ears is A where is the length of the compounding period (in ears). (a) Use a graphing utilit to graph A, where 0. (b) Use the zoom and trace features to estimate the balance for quarterl compounding and dail compounding. (c) Use the zoom and trace features to estimate 0 A. 0 p < 00 What do ou think this it represents Eplain our reasoning.