Infinite Limits. Let f be the function given by. f x 3 x 2.

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1 0_005.qd //0 :07 PM Page 8 SECTION.5 Infinite Limits 8, as Section.5, as + f() = f increases and decreases without bound as approaches. Figure.9 Infinite Limits Determine infinite its from the left and from the right. Find and sketch the vertical asmptotes of the graph of a function. Infinite Limits Let f be the function given b From Figure.9 and the table, ou can see that f decreases without bound as approaches from the left, and f increases without bound as approaches from the right. This behavior is denoted as and f. f decreases without bound as approaches from the left. f increases without bound as approaches from the right. approaches from the left. approaches from the right. f ? f decreases without bound. f increases without bound. A it in which f increases or decreases without bound as approaches c is called an infinite it. Definition of Infinite Limits Let f be a function that is defined at ever real number in some open interval containing c (ecept possibl at c itself). The statement M Infinite its Figure.0 δ c δ f() = c f c means that for each M > 0 there eists a > 0 such that f > M whenever (see Figure.0). Similarl, the statement 0 < c < f c means that for each N < 0 there eists a > 0 such that f < N whenever 0 < c To define the infinite it from the left, replace 0 < c <. < b c < < c. To define the infinite it from the right, replace 0 < c < b c < < c. Be sure ou see that the equal sign in the statement f does not mean that the it eists! On the contrar, it tells ou how the it fails to eist b denoting the unbounded behavior of f as approaches c.

2 0_005.qd //0 :07 PM Page 8 8 CHAPTER Limits and Their Properties EXPLORATION Use a graphing utilit to graph each function. For each function, analticall find the single real number c that is not in the domain. Then graphicall find the it of f as approaches c from the left and from the right. a. f b. f c. f d. f EXAMPLE Determining Infinite Limits from a Graph Use Figure. to determine the it of each function as approaches from the left and from the right. f() = f() = ( ) f() = f() = ( ) (a) (b) Figure. Each graph has an asmptote at. (c) (d) Solution a. and b. Limit from each side is. c. and d. Limit from each side is. Vertical Asmptotes If it were possible to etend the graphs in Figure. toward positive and negative infinit, ou would see that each graph becomes arbitraril close to the vertical line. This line is a vertical asmptote of the graph of f. (You will stud other tpes of asmptotes in Sections.5 and..) NOTE If the graph of a function f has a vertical asmptote at c, then f is not continuous at c. Definition of Vertical Asmptote If f approaches infinit (or negative infinit) as approaches c from the right or the left, then the line c is a vertical asmptote of the graph of f.

3 0_005.qd //0 :07 PM Page 85 SECTION.5 Infinite Limits 85 In Eample, note that each of the functions is a quotient and that the vertical asmptote occurs at a number where the denominator is 0 (and the numerator is not 0). The net theorem generalizes this observation. (A proof of this theorem is given in Appendi A.) THEOREM. Vertical Asmptotes Let f and g be continuous on an open interval containing c. If fc 0, gc 0, and there eists an open interval containing c such that g 0 for all c in the interval, then the graph of the function given b h f g has a vertical asmptote at c. f() = ( + ) (a) f() = + (b) π π π f() = cot (c) Functions with vertical asmptotes Figure. EXAMPLE Finding Vertical Asmptotes Determine all vertical asmptotes of the graph of each function. a. b. f f c. f cot Solution a. When, the denominator of f is 0 and the numerator is not 0. So, b Theorem., ou can conclude that is a vertical asmptote, as shown in Figure.(a). b. B factoring the denominator as f ou can see that the denominator is 0 at and. Moreover, because the numerator is not 0 at these two points, ou can appl Theorem. to conclude that the graph of f has two vertical asmptotes, as shown in Figure.(b). c. B writing the cotangent function in the form f cot cos sin ou can appl Theorem. to conclude that vertical asmptotes occur at all values of such that sin 0 and cos 0, as shown in Figure.(c). So, the graph of this function has infinitel man vertical asmptotes. These asmptotes occur when n, where n is an integer. Theorem. requires that the value of the numerator at c be nonzero. If both the numerator and the denominator are 0 at c, ou obtain the indeterminate form 00, and ou cannot determine the it behavior at c without further investigation, as illustrated in Eample.

4 0_005.qd //0 :07 PM Page 8 8 CHAPTER Limits and Their Properties EXAMPLE A Rational Function with Common Factors f() = + 8 Undefined when = Vertical asmptote at = f increases and decreases without bound as approaches. Figure. Determine all vertical asmptotes of the graph of Solution f 8. Begin b simplifing the epression, as shown. f 8, At all -values other than, the graph of f coincides with the graph of g. So, ou can appl Theorem. to g to conclude that there is a vertical asmptote at, as shown in Figure.. From the graph, ou can see that 8 and Note that is not a vertical asmptote. 8. EXAMPLE Determining Infinite Limits f() = Find each it. and Solution Because the denominator is 0 when (and the numerator is not zero), ou know that the graph of f has a vertical asmptote at. Figure. has a vertical asmptote at. This means that each of the given its is either or. A graphing utilit can help determine the result. From the graph of f shown in Figure., ou can see that the graph approaches from the left of and approaches from the right of. So, ou can conclude that and f. The it from the left is infinit. The it from the right is negative infinit. TECHNOLOGY PITFALL When using a graphing calculator or graphing software, be careful to interpret correctl the graph of a function with a vertical asmptote graphing utilities often have difficult drawing this tpe of graph.

5 0_005.qd //0 :07 PM Page 87 SECTION.5 Infinite Limits 87 THEOREM.5 Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that c and g L. c. Sum or difference: c. Product:, c L > 0, c L < 0. Quotient: g c f 0 Similar properties hold for one-sided its and for functions for which the it of f as approaches c is. Proof To show that the it of f g is infinite, choose M > 0. You then need to find > 0 such that f g > M whenever 0 < c <. For simplicit s sake, ou can assume L is positive. Let M M. Because the it of f is infinite, there eists such that f > M whenever 0 < c Also, because the it of is there eists such that g L <. g L, < whenever 0 < c B letting be the smaller of and ou can conclude that 0 < < c. implies f > M and, g L < <. The second of these two inequalities implies that g > L, and, adding this to the first inequalit, ou can write f g > M L M L > M. So, ou can conclude that f g. c The proofs of the remaining properties are left as eercises (see Eercise 7). EXAMPLE 5 Determining Limits a. Because and ou can write 0, 0 0. Propert, Theorem.5 b. Because and cot, ou can write 0. cot Propert, Theorem.5 c. Because and cot, ou can write 0 0 cot. 0 Propert, Theorem.5

6 0_005.qd //0 :07 PM Page CHAPTER Limits and Their Properties Eercises for Section.5 In Eercises, determine whether f approaches or as approaches from the left and from the right... f. f tan. Numerical and Graphical Analsis In Eercises 5 8, determine whether f approaches or as approaches from the left and from the right b completing the table. Use a graphing utilit to graph the function and confirm our answer. f f f f. f f f sec.00.5 f sec In Eercises 9 8, find the vertical asmptotes (if an) of the graph of the function. 9. f 0. f. h. g. f. f 5. s gt t. hs t s 5 7. f tan 8. f sec See for worked-out solutions to odd-numbered eercises. 9. Tt t 0. g. f. f 9 8. g. h 5.. ht t t f t g tan st t sin t In Eercises 9, determine whether the graph of the function has a vertical asmptote or a removable discontinuit at. Graph the function using a graphing utilit to confirm our answer f 7 f sin. f. f In Eercises 8, find the it sin. 5. csc In Eercises 9 5, use a graphing utilit to graph the function and determine the one-sided it. 9. f 50. f 5. f 5 5. f sec f f 5 0 cos 0 cot tan f f

7 0_005.qd //0 :07 PM Page 89 SECTION.5 Infinite Limits 89 Writing About Concepts 5. In our own words, describe the meaning of an infinite it. Is a real number? 5. In our own words, describe what is meant b an asmptote of a graph. 55. Write a rational function with vertical asmptotes at and, and with a zero at. 5. Does the graph of ever rational function have a vertical asmptote? Eplain. 57. Use the graph of the function f (see figure) to sketch the graph of g f on the interval,. To print an enlarged cop of the graph, go to the website f. Relativit According to the theor of relativit, the mass m of a particle depends on its velocit v. That is, m 0 m v c where m 0 is the mass when the particle is at rest and c is the speed of light. Find the it of the mass as v approaches c.. Rate of Change A 5-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled awa from the house at a rate of feet per second, the top will move down the wall at a rate of r ft/sec 5 where is the distance between the base of the ladder and the house. (a) Find the rate r when is 7 feet. (b) Find the rate r when is 5 feet. (c) Find the it of r as Bole s Law For a quantit of gas at a constant temperature, the pressure P is inversel proportional to the volume V. Find the it of P as V Rate of Change A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of revolution per second. The rate at which the light beam moves along the wall is r 50 sec ft/sec. (a) Find the rate r when (b) Find the rate r when (c) Find the it of r as is. is.. r 5 ft ft sec. Average Speed On a trip of d miles to another cit, a truck driver s average speed was miles per hour. On the return trip the average speed was miles per hour. The average speed for the round trip was 50 miles per hour. (a) Verif that 5 What is the domain? 5. (b) Complete the table. θ 50 ft Illegal Drugs The cost in millions of dollars for a governmental agenc to seize % of an illegal drug is C 58 00, 0 < 00. (a) Find the cost of seizing 5% of the drug. (b) Find the cost of seizing 50% of the drug. (c) Find the cost of seizing 75% of the drug. (d) Find the it of C as 00 and interpret its meaning. Are the values of different than ou epected? Eplain. (c) Find the it of as 5 and interpret its meaning.. Numerical and Graphical Analsis Use a graphing utilit to complete the table for each function and graph each function to estimate the it. What is the value of the it when the power on in the denominator is greater than? (a) (c) f sin 0 sin 0 sin (b) 0 sin (d) 0

8 0_005.qd //0 :07 PM Page CHAPTER Limits and Their Properties 5. Numerical and Graphical Analsis Consider the shaded region outside the sector of a circle of radius 0 meters and inside a right triangle (see figure). (a) Write the area A f of the region as a function of. Determine the domain of the function. (b) Use a graphing utilit to complete the table and graph the function over the appropriate domain. f (c) Find the it of A as θ Numerical and Graphical Reasoning A crossed belt connects a 0-centimeter pulle (0-cm radius) on an electric motor with a 0-centimeter pulle (0-cm radius) on a saw arbor (see figure). The electric motor runs at 700 revolutions per minute. 0 cm 0 cm (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let L be the total length of the belt. Write L as a function of, where is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulle.) 0 m. φ (d) Use a graphing utilit to complete the table. L (e) Use a graphing utilit to graph the function over the appropriate domain. (f) Find Use a geometric argument as the basis of a second method of finding this it. (g) Find True or False? In Eercises 7 70, 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 p is a polnomial, then the graph of the function given b f p has a vertical asmptote at. 8. The graph of a rational function has at least one vertical asmptote. 9. The graphs of polnomial functions have no vertical asmptotes. 70. If f has a vertical asmptote at 0, then f is undefined at Find functions f and g such that f and c g but f g 0. c c 7. Prove the remaining properties of Theorem Prove that if f, then 7. Prove that if 0, then f does not eist. c f c Infinite Limits In Eercises 75 and 7, use the - definition of infinite its to prove the statement L. L. 0 c c 0. f Section Project: Graphs and Limits of Trigonometric Functions Recall from Theorem.9 that the it of f sin as approaches 0 is. (a) Use a graphing utilit to graph the function f on the interval 0. Eplain how the graph helps confirm this theorem. (b) Eplain how ou could use a table of values to confirm the value of this it numericall. (c) Graph g sin b hand. Sketch a tangent line at the point 0, 0 and visuall estimate the slope of this tangent line. (d) Let, sin be a point on the graph of g near 0, 0, and write a formula for the slope of the secant line joining, sin and 0, 0. Evaluate this formula for 0. and 0.0. Then find the eact slope of the tangent line to g at the point 0, 0. (e) Sketch the graph of the cosine function h cos. What is the slope of the tangent line at the point 0,? Use its to find this slope analticall. (f) Find the slope of the tangent line to k tan at 0, 0.