L 8.6 L 10.6 L 13.3 L Psud = u 3-4u 2 + 5u; [1, 2]

Transcription

1 . Rates of Change and Tangents to Curves 6 p Q (5, ) (, ) (5, ) (, 65) Slope of PQ p/ t (flies/ da) L 8.6 L.6 L. L 6. Number of flies 5 B(5, 5) Q(5, ) 5 5 P(, 5) 5 5 A(, ) Time (das) t FIGURE.6 The positions and slopes of four secants through the point P on the fruit fl graph (Eample 5). The values in the table show that the secant slopes rise from 8.6 to 6. as the t-coordinate of Q decreases from 5 to, and we would epect the slopes to rise slightl higher as t continued on toward. Geometricall, the secants rotate about P and seem to approach the red tangent line in the figure. Since the line appears to pass through the points (, ) and (5, 5), it has slope = 6.7 flies>da (approimatel). On da the population was increasing at a rate of about 6.7 flies> da. The instantaneous rates in Eample were found to be the values of the average speeds, or average rates of change, as the time interval of length h approached. That is, the instantaneous rate is the value the average rate approaches as the length h of the interval over which the change occurs approaches zero. The average rate of change corresponds to the slope of a secant line; the instantaneous rate corresponds to the slope of the tangent line as the independent variable approaches a fied value. In Eample, the independent variable t approached the values t = and t =. In Eample, the independent variable approached the value =. So we see that instantaneous rates and slopes of tangent lines are closel connected. We investigate this connection thoroughl in the net chapter, but to do so we need the concept of a it. Eercises. Average Rates of Change In Eercises 6, find the average rate of change of the function over the given interval or intervals.. a. [, ] b. [-, ]. gsd =.. ƒsd = + a. [-, ] b. [-, ] hstd = cot t a. [p>, p>] b. [p>6, p>] gstd = + cos t a. [, p] b. [-p, p] 5. Rsud = u + ; [, ] 6. Slope of a Curve at a Point In Eercises 7, use the method in Eample to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P Psud = u - u + 5u; [, ] = -, P(, ) = 5 -, P(, ) = - -, P(, -) = -, P(, -). =, P(, 8)

2 6 Chapter : Limits and Continuit T... = -, P(, ) = -, P(, -) = - +, P(, ) Instantaneous Rates of Change 5. Speed of a car The accompaning figure shows the timeto-distance graph for a sports car accelerating from a standstill. Distance (m) a. Estimate the slopes of secants PQ, PQ, PQ, and PQ, arranging them in order in a table like the one in Figure.6. What are the appropriate units for these slopes? b. Then estimate the car s speed at time t = sec. 6. The accompaning figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 8 m to the surface of the moon. a. Estimate the slopes of the secants PQ, PQ, PQ, and PQ, arranging them in a table like the one in Figure.6. b. About how fast was the object going when it hit the surface? Distance fallen (m) s Q 5 Elapsed time (sec) 5 Elapsed time (sec) 7. The profits of a small compan for each of the first five ears of its operation are given in the following table: Q Year Profit in $s a. Plot points representing the profit as a function of ear, and join them b as smooth a curve as ou can. Q Q Q Q P Q Q t P t T T T b. What is the average rate of increase of the profits between and? c. Use our graph to estimate the rate at which the profits were changing in. 8. Make a table of values for the function Fsd = s + d>s - d at the points =., = >, = >, = >, = >, and =. a. Find the average rate of change of F() over the intervals [, ] for each Z in our table. b. Etending the table if necessar, tr to determine the rate of change of F() at =. 9. Let gsd = for Ú. a. Find the average rate of change of g() with respect to over the intervals [, ], [,.5] and [, + h]. b. Make a table of values of the average rate of change of g with respect to over the interval [, + h] for some values of h approaching zero, sa h =.,.,.,.,., and.. c. What does our table indicate is the rate of change of g() with respect to at =? d. Calculate the it as h approaches zero of the average rate of change of g() with respect to over the interval [, + h].. Let ƒstd = >t for t Z. a. Find the average rate of change of ƒ with respect to t over the intervals (i) from t = to t =, and (ii) from t = to t = T. b. Make a table of values of the average rate of change of ƒ with respect to t over the interval [, T], for some values of T approaching, sa T =.,.,.,.,., and.. c. What does our table indicate is the rate of change of ƒ with respect to t at t =? d. Calculate the it as T approaches of the average rate of change of ƒ with respect to t over the interval from to T. You will have to do some algebra before ou can substitute T =.. The accompaning graph shows the total distance s traveled b a bicclist after t hours. Distance traveled (mi) s Elapsed time (hr) a. Estimate the bicclist s average speed over the time intervals [, ], [,.5], and [.5,.5]. b. Estimate the bicclist s instantaneous speed at the times t =, t =, and t =. c. Estimate the bicclist s maimum speed and the specific time at which it occurs. t

3 . Limit of a Function and Limit Laws 65. The accompaning graph shows the total amount of gasoline A in the gas tank of an automobile after being driven for t das. Remaining amount (gal) 6 8 A Elapsed time (das) t a. Estimate the average rate of gasoline consumption over the time intervals [, ], [, 5], and [7, ]. b. Estimate the instantaneous rate of gasoline consumption at the times t =, t =, and t = 8. c. Estimate the maimum rate of gasoline consumption and the specific time at which it occurs.. Limit of a Function and Limit Laws In Section. we saw that its arise when finding the instantaneous rate of change of a function or the tangent to a curve. Here we begin with an informal definition of it and show how we can calculate the values of its. A precise definition is presented in the net section. HISTORICAL ESSAY Limits f() Limits of Function Values Frequentl when studing a function = ƒ(), we find ourselves interested in the function s behavior near a particular point, but not at. This might be the case, for instance, if is an irrational number, like p or, whose values can onl be approimated b close rational numbers at which we actuall evaluate the function instead. Another situation occurs when tring to evaluate a function at leads to division b zero, which is undefined. We encountered this last circumstance when seeking the instantaneous rate of change in b considering the quotient function >h for h closer and closer to zero. Here s a specific eample where we eplore numericall how a function behaves near a particular point at which we cannot directl evaluate the function. EXAMPLE How does the function behave near =? ƒsd = - - Solution The given formula defines ƒ for all real numbers ecept = (we cannot divide b zero). For an Z, we can simplif the formula b factoring the numerator and canceling common factors: ƒsd = s - ds + d - = + for Z. FIGURE.7 The graph of ƒ is identical with the line = + ecept at =, where ƒ is not defined (Eample ). The graph of ƒ is the line = + with the point (, ) removed. This removed point is shown as a hole in Figure.7. Even though ƒ() is not defined, it is clear that we can make the value of ƒ() as close as we want to b choosing close enough to (Table.).

4 . Limit of a Function and Limit Laws 7 Another important propert of its is given b the net theorem. A proof is given in the net section. THEOREM 5 If ƒsd gsd for all in some open interval containing c,ecept possibl at = c itself, and the its of ƒ and g both eist as approaches c, then ƒsd gsd. :c :c The assertion resulting from replacing the less than or equal to ( ) inequalit b the strict less than (6) inequalit in Theorem 5 is false. Figure.a shows that for u Z, -ƒ u ƒ 6 sin u 6 ƒ u ƒ, but in the it as u :, equalit holds. Eercises. Limits from Graphs. For the function g() graphed here, find the following its or eplain wh the do not eist. a. gsd b. gsd c. gsd d. : : :. For the function ƒ(t) graphed here, find the following its or eplain wh the do not eist. a. ƒstd b. ƒstd c. ƒstd d. t: - t: - t: s f(t) g() s t gsd :.5 ƒstd t: -.5. Which of the following statements about the function graphed here are true, and which are false? a. ƒsd does not eist. : b. ƒsd = : c. ƒsd does not eist. : d. ƒsd eists at ever point in s -, d. : e. ƒsd eists at ever point in (, ). : f() f() = ƒsd. Which of the following statements about the function graphed here are true, and which are false? a. ƒsd eists. : b. ƒsd = : c. ƒsd = : d. ƒsd = : e. ƒsd = : f. ƒsd eists at ever point in s -, d. : g. ƒsd does not eist. : = ƒsd Eistence of Limits In Eercises 5 and 6, eplain wh the its do not eist : ƒ ƒ : - 7. Suppose that a function ƒ() is defined for all real values of ecept =. Can anthing be said about the eistence of : ƒsd? Give reasons for our answer. 8. Suppose that a function ƒ() is defined for all in [-, ]. Can anthing be said about the eistence of : ƒsd? Give reasons for our answer.

5 7 Chapter : Limits and Continuit 9. If : ƒsd = 5, must ƒ be defined at =? If it is, must ƒsd = 5? Can we conclude anthing about the values of ƒ at =? Eplain.. If ƒsd = 5, must : ƒsd eist? If it does, then must : ƒsd = 5? Can we conclude anthing about : ƒsd? Eplain. Calculating Limits Find the its in Eercises.. s + 5d. : -7. 8st - 5dst - 7d. t: : 7. s - d 8. : - 9. s5 - d>. : -.. h: h + + Limits of quotients Find the its in Eercises : : -5 t 7. + t - 8. t: t : : - u. -. u: u : : : : - Limits with trigonometric functions 5.. ( sin - ). : 5. sec 6. : + + sin cos : s : d : - s d sss - d s:> : sz - 8d> z: 5h + - h: h : : - t + t + t: - t - t : : : : : - + : - : Find the its in Eercises : sin tan : : ( - )( - cos ) 9. + cos ( + p) sec : -p : Using Limit Rules 5. Suppose : ƒsd = and : gsd = -5. Name the rules in Theorem that are used to accomplish steps (a), (b), and (c) of the following calculation. ƒsd - gsd : sƒsd + 7d = > 5. Let : hsd = 5, : psd =, and : rsd =. Name the rules in Theorem that are used to accomplish steps (a), (b), and (c) of the following calculation. : 5. Suppose :c ƒsd = 5 and :c gsd = -. Find a. ƒsdgsd b. :c c. sƒsd + gsdd d. :c 5. Suppose : ƒsd = and : gsd = -. Find a. sgsd + d b. : c. d. : sgsdd 55. Suppose :b ƒsd = 7 and :b gsd = -. Find a. sƒsd + gsdd b. :b c. gsd d. :b 56. Suppose that :- psd =, :- rsd =, and :- ssd = -. Find a. spsd + rsd + ssdd : - b. : - psd # rsd # ssd c. s -psd + 5rsdd>ssd : - = = = = = 5hsd psds - rsdd = sƒsd - gsdd : sƒsd + 7d> : ƒsd - gsd : : A Aƒsd + 7BB > : ƒsd - gsd : : A ƒ() + 7B > : : sdsd - s -5d s + 7d > = 7 : 5hsd : spsds - rsddd 5hsd : A p()ba A - r()bb : : 5 hsd : A p()ba - r()b : : : = s5ds5d sds - d = 5 :c ƒsdgsd :c ƒsd : : ƒsd ƒsd - gsd gsd ƒsd - ƒsd # gsd :b ƒsd>gsd :b (a) (b) (c) (a) (b) (c)

6 . Limit of a Function and Limit Laws 75 Limits of Average Rates of Change Because of their connection with secant lines, tangents, and instantaneous rates, its of the form occur frequentl in calculus. In Eercises 57 6, evaluate this it for the given value of and function ƒ. 57. ƒsd =, = 58. ƒsd =, = ƒsd = -, = 6. ƒsd = >, = - 6. ƒsd =, = 7 6. ƒsd = +, = Using the Sandwich Theorem ƒs + hd - ƒsd h: h 6. If 5 - ƒsd 5 - for -, find : ƒsd. 6. If - gsd cos for all, find : gsd. 65. a. It can be shown that the inequalities sin - cos 6 hold for all values of close to zero. What, if anthing, does this tell ou about sin : - cos? Give reasons for our answer. T b. Graph = - s >6d, = s sin d>s - cos d, and = together for -. Comment on the behavior of the graphs as :. 66. a. Suppose that the inequalities cos 6 hold for values of close to zero. (The do, as ou will see in Section.9.) What, if anthing, does this tell ou about - cos :? Give reasons for our answer. T b. Graph the equations = s>d - s >d, = s - cos d>, and = > together for -. Comment on the behavior of the graphs as :. Estimating Limits T You will f ind a graphing calculator useful for Eercises Let ƒsd = s - 9d>s + d. a. Make a table of the values of ƒ at the points = -., -., -., and so on as far as our calculator can go. Then estimate :- ƒsd. What estimate do ou arrive at if ou evaluate ƒ at = -.9, -.99, -.999, Á instead? b. Support our conclusions in part (a) b graphing ƒ near = - and using Zoom and Trace to estimate -values on the graph as : -. c. Find :- ƒsd algebraicall, as in Eample Let gsd = s - d>( - ). a. Make a table of the values of g at the points =.,.,., and so on through successive decimal approimations of. Estimate : gsd. b. Support our conclusion in part (a) b graphing g near = and using Zoom and Trace to estimate -values on the graph as :. c. Find : gsd algebraicall. 69. Let Gsd = s + 6d>s + - d. a. Make a table of the values of G at = -5.9, -5.99, , and so on. Then estimate :-6 Gsd. What estimate do ou arrive at if ou evaluate G at = -6., -6., -6., Á instead? b. Support our conclusions in part (a) b graphing G and using Zoom and Trace to estimate -values on the graph as : -6. c. Find :-6 Gsd algebraicall. 7. Let hsd = s - - d>s - + d. a. Make a table of the values of h at =.9,.99,.999, and so on. Then estimate : hsd. What estimate do ou arrive at if ou evaluate h at =.,.,., Á instead? b. Support our conclusions in part (a) b graphing h near = and using Zoom and Trace to estimate -values on the graph as :. c. Find : hsd algebraicall. 7. Let ƒsd = s - d>s ƒ ƒ - d. a. Make tables of the values of ƒ at values of that approach = - from above and below. Then estimate :- ƒsd. b. Support our conclusion in part (a) b graphing ƒ near = - and using Zoom and Trace to estimate -values on the graph as : -. c. Find :- ƒsd algebraicall. 7. Let Fsd = s + + d>s - ƒ ƒ d. a. Make tables of values of F at values of that approach = - from above and below. Then estimate :- Fsd. b. Support our conclusion in part (a) b graphing F near = - and using Zoom and Trace to estimate -values on the graph as : -. c. Find :- Fsd algebraicall. 7. Let gsud = ssin ud>u. a. Make a table of the values of g at values of u that approach u = from above and below. Then estimate u: gsud. b. Support our conclusion in part (a) b graphing g near u =. 7. Let Gstd = s - cos td>t. a. Make tables of values of G at values of t that approach t = from above and below. Then estimate t: Gstd. b. Support our conclusion in part (a) b graphing G near t =. 75. Let ƒsd = >s - d. a. Make tables of values of ƒ at values of that approach = from above and below. Does ƒ appear to have a it as :? If so, what is it? If not, wh not? b. Support our conclusions in part (a) b graphing ƒ near =.

7 76 Chapter : Limits and Continuit 76. Let ƒsd = s - d>. a. Make tables of values of ƒ at values of that approach = from above and below. Does ƒ appear to have a it as :? If so, what is it? If not, wh not? b. Support our conclusions in part (a) b graphing ƒ near =. Theor and Eamples 77. If ƒsd for in [-, ] and ƒsd for 6 - and 7, at what points c do ou automaticall know :c ƒsd? What can ou sa about the value of the it at these points? 78. Suppose that gsd ƒsd hsd for all Z and suppose that gsd = hsd = -5. : : Can we conclude anthing about the values of ƒ, g, and h at =? Could ƒsd =? Could : ƒsd =? Give reasons for our answers. 79. If ƒsd - 5 =, find ƒsd. : - : ƒsd 8. If find : - =, a. b. : - ƒsd ƒsd : - ƒsd a. If =, find ƒsd. : - : T T b. If ƒsd - 5 =, find ƒsd. : - : ƒsd 8. If find : =, a. ƒsd b. : 8. a. Graph gsd = sin s>d to estimate : gsd, zooming in on the origin as necessar. b. Confirm our estimate in part (a) with a proof. 8. a. Graph hsd = cos s> d to estimate : hsd, zooming in on the origin as necessar. b. Confirm our estimate in part (a) with a proof. COMPUTER EXPLORATIONS Graphical Estimates of Limits In Eercises 85 9, use a CAS to perform the following steps: a. Plot the function near the point being approached. b. From our plot guess the value of the it : ƒsd : : - s + d : : cos 9. : sin : - cos. 9 To satisf this 7 5 The Precise Definition of a Limit 5 Restrict to this Upper bound: 9 Lower bound: 5 FIGURE.5 Keeping within unit of = will keep within units of = 7 (Eample ). We now turn our attention to the precise definition of a it. We replace vague phrases like gets arbitraril close to in the informal definition with specific conditions that can be applied to an particular eample. With a precise definition, we can prove the it properties given in the preceding section and establish man important its. To show that the it of ƒ() as : equals the number L, we need to show that the gap between ƒ() and L can be made as small as we choose if is kept close enough to. Let us see what this would require if we specified the size of the gap between ƒ() and L. EXAMPLE Consider the function = - near =. Intuitivel it appears that is close to 7 when is close to, so : s - d = 7. However, how close to = does have to be so that = - differs from 7 b, sa, less than units? Solution We are asked: For what values of is ƒ - 7 ƒ 6? To find the answer we first epress ƒ - 7 ƒ in terms of : ƒ - 7 ƒ = ƒ s - d - 7 ƒ = ƒ - 8 ƒ. The question then becomes: what values of satisf the inequalit find out, we solve the inequalit: ƒ - 8 ƒ ƒ - 8 ƒ 6? To Keeping within unit of = will keep within units of = 7 (Figure.5).

8 8 Chapter : Limits and Continuit Therefore, for an P7, there eists d 7 such that ƒ sgsd - ƒsdd - sm - Ld ƒ 6P whenever 6 ƒ - c ƒ 6 d. Since L - M 7 b hpothesis, we take P=L - M in particular and we have a number d 7 such that ƒ sgsd - ƒsdd - sm - Ld ƒ 6 L - M whenever 6 ƒ - c ƒ 6 d. Since a ƒ a ƒ for an number a, we have sgsd - ƒsdd - sm - Ld 6 L - M whenever 6 ƒ - c ƒ 6 d which simplifies to gsd 6 ƒsd whenever 6 ƒ - c ƒ 6 d. But this contradicts ƒsd gsd. Thus the inequalit L 7 M must be false. Therefore L M. Eercises. Centering Intervals About a Point In Eercises 6, sketch the interval (a, b) on the -ais with the point inside. Then find a value of d 7 such that for all, 6 ƒ - ƒ 6 d Q a 6 6 b Finding Deltas Graphicall In Eercises 7, use the graphs to find a d 7 such that for all 6 ƒ - ƒ 6 d Q ƒ ƒsd - L ƒ 6P a =, b = 7, = 5 a =, b = 7, = a = -7>, b = ->, = - a = -7>, b = ->, = -> a = >9, b = >7, = > a =.759, b =.9, = NOT TO SCALE f() 5 L 6. f() L f() L f() L NOT TO SCALE f() L.5 f() L..6. NOT TO SCALE NOT TO SCALE NOT TO SCALE

9 . The Precise Definition of a Limit 8.. f() L f() L. Using the Formal Definition Each of Eercises 6 gives a function ƒ(), a point, and a positive number P. Find L = ƒsd. Then find a number d 7 such : that for all ƒ - ƒ 6 d Q ƒƒsd - L ƒ 6P. ƒsd = -, =, P =. ƒsd = - -, = -, P =. ƒsd = - -, =, P =.5 ƒsd = , + 5 = -5, P =.5 5. ƒsd = - 5, = -, P =.5 Finding Deltas Algebraicall Each of Eercises 5 gives a function ƒ() and numbers L,, and P7. In each case, find an open interval about on which the inequalit ƒ ƒsd - L ƒ 6Pholds. Then give a value for d 7 such that for all satisfing 6 ƒ - ƒ 6 d the inequalit ƒ ƒsd - L ƒ 6P holds ƒsd = +, L = 5, =, P =. ƒsd = -, L = -6, = -, P =. ƒsd = +, L =, =, P =. ƒsd =, L = >, = >, P =. ƒsd = 9 -, L =, =, P = ƒsd = - 7, L =, =, P = ƒsd = >, L = >, =, P = NOT TO SCALE 6. Prove the it statements in Eercises s9 - d = =. :9... = = ƒsd = >, =, P =. : : : - s - 7d = : - = : ƒsd = if ƒsd = e, Z :, = ƒsd = if ƒsd = e, Z - : -, = - : = - : - = -, 6 ƒsd = if ƒsd = e : 6 -, Ú, 6 ƒsd = if ƒsd = e : >, Ú... ƒsd =, L =, =, P =. ƒsd =, L =, = -, P =.5 ƒsd = >, L = -, = -, P =. 9. : sin = 5. ƒsd = - 5, L =, =, P = ƒsd = >, L = 5, =, P = ƒsd = m, m 7, L = m, =, P=. ƒsd = m, m 7, L = m, =, ƒsd = m + b, m 7, L = sm>d + b, = >, P =c 7 ƒsd = m + b, m 7, L = m + b, =, P=.5 P=c 7 sin

10 8 Chapter : Limits and Continuit 5. : sin = When Is a Number L Not the Limit of ƒ() as :? Showing L is not a it We can prove that : ƒsd Z L b providing an P7such that no possible d 7 satisfies the condition for all, 6 ƒ - ƒ 6 d Q ƒƒsd - L ƒ 6P. We accomplish this for our candidate P d 7 there eists a value of such that b showing that for each sin 6 ƒ - ƒ 6 d and ƒƒsd - L ƒ ÚP. f() L L Theor and Eamples 5. Define what it means to sa that gsd = k. : 5. Prove that ƒsd = L if and onl if ƒsh + cd = L. :c h: 5. A wrong statement about its Show b eample that the following statement is wrong. The number L is the it of ƒ() as approaches if ƒ() gets closer to L as approaches. Eplain wh the function in our eample does not have the given value of L as a it as :. 5. Another wrong statement about its Show b eample that the following statement is wrong. The number L is the it of ƒ() as approaches if, given an P7, there eists a value of for which ƒ ƒsd - L ƒ 6P. 57. L f() Let ƒsd = e, 6 +, 7. a value of for which and f() L T Eplain wh the function in our eample does not have the given value of L as a it as :. 55. Grinding engine clinders Before contracting to grind engine clinders to a cross-sectional area of 9 in, ou need to know how much deviation from the ideal clinder diameter of =.85 in. ou can allow and still have the area come within. in of the required 9 in. To find out, ou let A = ps>d and look for the interval in which ou must hold to make ƒ A - 9 ƒ.. What interval do ou find? 56. Manufacturing electrical resistors Ohm s law for electrical circuits like the one shown in the accompaning figure states that V = RI. In this equation, V is a constant voltage, I is the current in amperes, and R is the resistance in ohms. Your firm has been asked to suppl the resistors for a circuit in which V will be volts and I is to be 5 ;. amp. In what interval does R have to lie for I to be within. amp of the value I = 5? f() a. Let P=>. Show that no possible d 7 satisfies the following condition: For all, 6 ƒ - ƒ 6 d Q ƒ ƒsd - ƒ 6 >. That is, for each d 7 show that there is a value of such that 6 ƒ - ƒ 6 d and ƒƒsd - ƒ Ú >. V I R This will show that : ƒsd Z. b. Show that : ƒsd Z. c. Show that : ƒsd Z.5.

11 . One-Sided Limits , 6 Let hsd =, =, 7. h() g() Show that a. hsd Z : b. hsd Z : c. hsd Z : 59. For the function graphed here, eplain wh a. b. c. ƒsd Z : ƒsd Z.8 : ƒsd Z :.8 f() 6. a. For the function graphed here, show that : - gsd Z. b. Does : - gsd appear to eist? If so, what is the value of the it? If not, wh not? COMPUTER EXPLORATIONS In Eercises 6 66, ou will further eplore finding deltas graphicall. Use a CAS to perform the following steps: a. Plot the function = ƒsd near the point being approached. b. Guess the value of the it L and then evaluate the it smbolicall to see if ou guessed correctl. c. Using the value P=., graph the banding lines = L -P and = L +Ptogether with the function ƒ near. d. From our graph in part (c), estimate a d 7 such that for all 6 ƒ - ƒ 6 d Q ƒƒsd - L ƒ 6P. Test our estimate b plotting ƒ,, and over the interval 6 ƒ - ƒ 6 d. For our viewing window use - d + d and L - P L + P. If an function values lie outside the interval [L -P, L +P], our choice of d was too large. Tr again with a smaller estimate. e. Repeat parts (c) and (d) successivel for P=.,.5, and.. 6. ƒsd = - 8 -, = ƒsd = , = ƒsd = ƒsd = sin, = s - cos d - sin, = ƒsd = - -, = 66. ƒsd = - s7 + d + 5, - =. One-Sided Limits In this section we etend the it concept to one-sided its, which are its as approaches the number c from the left-hand side (where 6 c) or the right-hand side s 7 cd onl. One-Sided Limits To have a it L as approaches c, a function ƒ must be defined on both sides of c and its values ƒ() must approach L as approaches c from either side. Because of this, ordinar its are called two-sided.

12 9 Chapter : Limits and Continuit Solution (a) Using the half-angle formula cos h = - sin sh>d, we calculate cos h - h: h = - sin sh>d h: h sin u = - u: u sin u = -sdsd =. Let u = h>. Eq. () and Eample a in Section. (b) Equation () does not appl to the original fraction. We need a in the denominator, not a 5. We produce it b multipling numerator and denominator b >5: sin : 5 = : s>5d # sin s>5d # 5 = 5 sin : Now, Eq. () applies with u =. tan t sec t EXAMPLE 6 Find. t t: = 5 sd = 5 Solution From the definition of tan t and sec t, we have tan t sec t t: t = sin t # # t: t cos t cos t = ()()() =. Eq. () and Eample b in Section. Eercises. Finding Limits Graphicall. Which of the following statements about the function graphed here are true, and which are false? f() = ƒsd f() a. : - + b. : - c. : - d. ƒsd = - : : + e. ƒsd eists. : f. : g. : h. : i. : j. : - k. ƒsd does not eist. - : - l. : +. Which of the following statements about the function = ƒsd graphed here are true, and which are false? a. ƒsd = b. ƒsd does not eist. : - + : c. : d. : - e. : + f. does not eist. : g. ƒsd = + : : - h. eists at ever c in the open interval s -, d. :c i. eists at ever c in the open interval (, ). :c j. : - - k. does not eist. : +

13 . One-Sided Limits 9. Let -, 6 ƒsd = +, Let gsd = sins>d. sin a. Find : + ƒsd and : - ƒsd. b. Does : ƒsd eist? If so, what is it? If not, wh not? c. Find : - ƒsd and : + ƒsd. d. Does : ƒsd eist? If so, what is it? If not, wh not? -, 6. Let ƒsd =, = d, 7. a. Find : + ƒsd, : - ƒsd, and ƒ(). b. Does : ƒsd eist? If so, what is it? If not, wh not? c. Find :- - ƒsd and :- + ƒsd. d. Does :- ƒsd eist? If so, what is it? If not, wh not? 5. Let, ƒsd = sin, 7. a. Does : + ƒsd eist? If so, what is it? If not, wh not? b. Does : - ƒsd eist? If so, what is it? If not, wh not? c. Does : ƒsd eist? If so, what is it? If not, wh not?, sin, a. Does : + gsd eist? If so, what is it? If not, wh not? b. Does : - gsd eist? If so, what is it? If not, wh not? c. Does : gsd eist? If so, what is it? If not, wh not? 7. a. Graph b. Find : - ƒsd and : + ƒsd. c. Does : ƒsd eist? If so, what is it? If not, wh not? 8. a. Graph b. Find : + ƒsd and : - ƒsd. c. Does : ƒsd eist? If so, what is it? If not, wh not? Graph the functions in Eercises 9 and. Then answer these questions. 9.. a. What are the domain and range of ƒ? b. At what points c, if an, does :c ƒsd eist? c. At what points does onl the left-hand it eist? d. At what points does onl the right-hand it eist? Finding One-Sided Limits Algebraicall Find the its in Eercises , 6 ƒsd =, 6, =, - 6, or 6 ƒsd =, =, 6 - or 7 : A a : ba b a : - ƒsd = e, Z, =. ƒsd = e -, Z, =. + ba + 6 ba - b 7 h + h h: + h : + A - +

14 9 Chapter : Limits and Continuit h + h + 6 h: - h ƒ + ƒ 7. a. +s + d b. + s - d 8. a. b. : + ƒ - ƒ Use the graph of the greatest integer function = :;, Figure. in Section., to help ou find the its in Eercises 9 and. :u; :u; 9. a. b. u: + u u: - u. a. +st - :t; d b. sin U Using U: U Find the its in Eercises.. sinu u: u.. sin. : tan : csc cos 5 : + cos 9.. sin cos : - cos u.. sin u u: sin s - cos td.. - cos t t: sin u sin u u: : - t: ƒ + ƒ : - -s + d + s - d : - ƒ - ƒ t: -st - :t; d sin kt t: t h h: - sin h t: t tan t : 6 scot dscsc d - + sin : - cos : sin sin ssin hd h: sin h sin 5 : sin 7. u cos u 8. sin u cot u u: u: tan sin cot : sin 8 : cot sk constantd tan u.. u: u cot u Theor and Eamples. Once ou know :a + ƒsd and :a - ƒsd at an interior point of the domain of ƒ, do ou then know :a ƒsd? Give reasons for our answer.. If ou know that :c ƒsd eists, can ou find its value b calculating :c + ƒsd? Give reasons for our answer. 5. Suppose that ƒ is an odd function of. Does knowing that : + ƒsd = tell ou anthing about : - ƒsd? Give reasons for our answer. 6. Suppose that ƒ is an even function of. Does knowing that : - ƒsd = 7 tell ou anthing about either :- - ƒsd or :- + ƒsd? Give reasons for our answer. Formal Definitions of One-Sided Limits 7. Given P7, find an interval I = s5, 5 + dd, d 7, such that if lies in I, then - 5 6P. What it is being verified and what is its value? 8. Given P7, find an interval I = s - d, d, d 7, such that if lies in I, then - 6P. What it is being verified and what is its value? Use the definitions of right-hand and left-hand its to prove the it statements in Eercises 9 and = - 5. = : - ƒ ƒ : + ƒ - ƒ 5. Greatest integer function Find (a) : + :; and (b) : - :; ; then use it definitions to verif our findings. (c) Based on our conclusions in parts (a) and (b), can ou sa anthing about : :;? Give reasons for our answer. 5. One-sided its Let u: u cot u sin u cot u ƒsd = e sin s>d, 6, 7. Find (a) : + ƒsd and (b) : - ƒsd; then use it definitions to verif our findings. (c) Based on our conclusions in parts (a) and (b), can ou sa anthing about : ƒsd? Give reasons for our answer..5 Continuit When we plot function values generated in a laborator or collected in the field, we often connect the plotted points with an unbroken curve to show what the function s values are likel to have been at the times we did not measure (Figure.). In doing so, we are assuming that we are working with a continuous function, so its outputs var continuousl with the inputs and do not jump from one value to another without taking on the values in between. The it of a continuous function as approaches c can be found simpl b calculating the value of the function at c. (We found this to be true for polnomials in Theorem.) Intuitivel, an function = ƒsd whose graph can be sketched over its domain in one continuous motion without lifting the pencil is an eample of a continuous function. In this section we investigate more precisel what it means for a function to be continuous.

15 .5 Continuit function = + 5. Then ƒ is the sum of the function g and the quadratic function =, and the quadratic function is continuous for all values of. It follows that ƒ() = is continuous on the interval [-5>, q). B trial and error, we find the function values ƒ() = 5 L. and ƒ() = 9 + = 7, and note that ƒ is also continuous on the finite closed interval [, ] ( [-5>, q). Since the value = is between the numbers. and 7, b the Intermediate Value Theorem there is a number c H [, ] such that ƒ(c) =. That is, the number c solves the original equation. Eercises.5 Continuit from Graphs In Eercises, sa whether the function graphed is continuous on [-, ]. If not, where does it fail to be continuous and wh?.... f() h() Eercises 5 refer to the function ƒsd = e graphed in the accompaning figure. f() -, - 6, 6 6, = - +, 6 6, 6 6 (, ) (, ) g() k() 5. a. Does ƒs -d eist? b. Does : - + ƒsd eist? c. Does :- + ƒsd = ƒs -d? d. Is ƒ continuous at = -? 6. a. Does ƒ() eist? b. Does : ƒsd eist? c. Does : ƒsd = ƒsd? d. Is ƒ continuous at =? 7. a. Is ƒ defined at =? (Look at the definition of ƒ.) b. Is ƒ continuous at =? 8. At what values of is ƒ continuous? 9. What value should be assigned to ƒ() to make the etended function continuous at =?. To what new value should ƒ() be changed to remove the discontinuit? Appling the Continuit Test At which points do the functions in Eercises and fail to be continuous? At which points, if an, are the discontinuities removable? Not removable? Give reasons for our answers.. Eercise, Section.. Eercise, Section. At what points are the functions in Eercises continuous?. =. = = 6. = = ƒ - ƒ + sin 8. = 9. = cos.. = csc.. = tan. + s + d ƒ ƒ + - = cos + = tan p = + + sin 5. = + 6. = - The graph for Eercises = s - d > 8. = s - d >5

16 Chapter : Limits and Continuit 9.. Limits Involving Trigonometric Functions Find the its in Eercises 8. Are the functions continuous at the point being approached?. sin s - sin d , gsd = - Z 5, = - 8, Z, Z - - ƒsd = d, =, = - :p sec s : sec - tan - d : tan ap cos ssin > db 5. cos a p 6. t: 9 - sec t b 7. sin ap 8. : + e b Continuous Etensions 9. Define g() in a wa that etends gsd = s - 9d>s - d to be continuous at =.. Define h() in a wa that etends hstd = st + t - d>st - d to be continuous at t =.. Define ƒ() in a wa that etends ƒssd = ss - d>ss - d to be continuous at s =.. Define g() in a wa that etends to be continuous at =.. For what value of a is continuous at ever?. For what value of b is continuous at ever? 5. For what values of a is continuous at ever? 6. For what value of b is continuous at ever? : cos- (ln ) gsd = s - 6d>s - - d ƒsd = e -, 6 a, Ú, 6 - gsd = e b, Ú - ƒsd = b a - a, Ú, 6 - b gsd = b +, 6 + b, 7 sin ap cos stan tdb t: :p/6 csc + 5 tan 7. For what values of a and b is continuous at ever? 8. For what values of a and b is continuous at ever? -, - ƒsd = a - b, - 6 6, Ú a + b, gsd = + a - b, 6-5, 7 T In Eercises 9 5, graph the function ƒ to see whether it appears to have a continuous etension to the origin. If it does, use Trace and Zoom to find a good candidate for the etended function s value at =. If the function does not appear to have a continuous etension, can it be etended to be continuous at the origin from the right or from the left? If so, what do ou think the etended function s value(s) should be? 9. ƒsd = ƒsd = sin 5. ƒsd = s + d > ƒ ƒ Theor and Eamples 5. A continuous function = ƒsd is known to be negative at = and positive at =. Wh does the equation ƒsd = have at least one solution between = and =? Illustrate with a sketch. 5. Eplain wh the equation cos = has at least one solution. 55. Roots of a cubic Show that the equation = has three solutions in the interval [-, ]. 56. A function value Show that the function Fsd = s - ad # s - bd + takes on the value sa + bd> for some value of. 57. Solving an equation If ƒsd = - 8 +, show that there are values c for which ƒ(c) equals (a) p; (b) - ; (c) 5,,. 58. Eplain wh the following five statements ask for the same information. a. Find the roots of ƒsd = - -. b. Find the -coordinates of the points where the curve = crosses the line = +. c. Find all the values of for which - =. d. Find the -coordinates of the points where the cubic curve = - crosses the line =. e. Solve the equation - - =. ƒsd = ƒ ƒ Removable discontinuit Give an eample of a function ƒ() that is continuous for all values of ecept =, where it has a removable discontinuit. Eplain how ou know that ƒ is discontinuous at =, and how ou know the discontinuit is removable. 6. Nonremovable discontinuit Give an eample of a function g() that is continuous for all values of ecept = -, where it has a nonremovable discontinuit. Eplain how ou know that g is discontinuous there and wh the discontinuit is not removable.

17 .6 Limits Involving Infinit; Asmptotes of Graphs 6. A function discontinuous at ever point a. Use the fact that ever nonempt interval of real numbers contains both rational and irrational numbers to show that the function ƒsd = e, if is rational, if is irrational is discontinuous at ever point. b. Is ƒ right-continuous or left-continuous at an point? 6. If functions ƒ() and g() are continuous for, could ƒ()>g() possibl be discontinuous at a point of [, ]? Give reasons for our answer. 6. If the product function hsd = ƒsd # gsd is continuous at =, must ƒ() and g() be continuous at =? Give reasons for our answer. 6. Discontinuous composite of continuous functions Give an eample of functions ƒ and g, both continuous at =, for which the composite ƒ g is discontinuous at =. Does this contradict Theorem 9? Give reasons for our answer. 65. Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for our answer. 66. Stretching a rubber band Is it true that if ou stretch a rubber band b moving one end to the right and the other to the left, some point of the band will end up in its original position? Give reasons for our answer. 67. A fied point theorem Suppose that a function ƒ is continuous on the closed interval [, ] and that ƒsd for ever in [, ]. Show that there must eist a number c in [, ] such that ƒscd = c (c is called a fied point of ƒ). T 68. The sign-preserving propert of continuous functions Let ƒ be defined on an interval (a, b) and suppose that ƒscd Z at some c where ƒ is continuous. Show that there is an interval sc - d, c + dd about c where ƒ has the same sign as ƒ(c). 69. Prove that ƒ is continuous at c if and onl if 7. Use Eercise 69 together with the identities to prove that both ƒsd = sin and gsd = cos are continuous at ever point = c. Solving Equations Graphicall Use the Intermediate Value Theorem in Eercises 7 78 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations = = s - d = sone rootd = sin sh + cd = sin h cos c + cos h sin c, cos sh + cd = cos h cos c - sin h sin c + + = ƒsc + hd = ƒscd. h: = sthree rootsd 77. cos = sone rootd. Make sure ou are using radian mode. 78. sin = sthree rootsd. Make sure ou are using radian mode..6 Limits Involving Infinit; Asmptotes of Graphs FIGURE.9 The graph of = > approaches as : q or : - q. In this section we investigate the behavior of a function when the magnitude of the independent variable becomes increasingl large, or : ;q. We further etend the concept of it to infinite its, which are not its as before, but rather a new use of the term it. Infinite its provide useful smbols and language for describing the behavior of functions whose values become arbitraril large in magnitude. We use these it ideas to analze the graphs of functions having horizontal or vertical asmptotes. Finite Limits as : ˆ The smbol for infinit s q d does not represent a real number. We use q to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For eample, the function ƒsd = > is defined for all Z (Figure.9). When is positive and becomes increasingl large, > becomes increasingl small. When is negative and its magnitude becomes increasingl large, > again becomes small. We summarize these observations b saing that ƒsd = > has it as : q or : - q, or that is a it of ƒsd = > at infinit and negative infinit. Here are precise definitions.

18 Chapter : Limits and Continuit However, calculating more complicated its involving transcendental functions such as : e -, : ln, and a + : b requires more than simple algebraic techniques. The derivative is eactl the tool we need to calculate its in these kinds of cases (see Section.5), and this notion is the main subject of our net chapter. Eercises.6 Finding Limits. For the function ƒ whose graph is given, determine the following its. a. ƒ() b. ƒ() c. ƒ() : d. ƒ() : e. - + ƒ() : - - f. ƒ() : g. - ƒ() : + h. ƒ() : - i. ƒ() : : q : -q 6 5. For the function ƒ whose graph is given, determine the following its. a. ƒ() b. ƒ() c. ƒ() : d. ƒ() : + e. ƒ() : - f. ƒ() : g. ƒ() : h. - + ƒ() : - - i. ƒ() : j. - ƒ() : + k. ƒ() : - l. ƒ() : : q : -q 6 5 In Eercises 8, find the it of each function (a) as : q and (b) as : - q. (You ma wish to visualize our answer with a graphing calculator or computer.). ƒsd = -. ƒsd = p f 5 6 f 5. gsd = 6. gsd = + s>d -5 + s7>d - s>d 7. hsd = 8. hsd = - s> d + (> ) Find the its in Eercises 9. sin 9.. : q - t + sin t.. t + cos t t: -q Limits of Rational Functions In Eercises, find the it of each rational function (a) as : q and (b) as : - q.. ƒsd = +. ƒsd = ƒsd = hsd = 8. gsd = gsd = hsd =. hsd = hsd = Limits as : ˆ or : ˆ The process b which we determine its of rational functions applies equall well to ratios containing noninteger or negative powers of : divide numerator and denominator b the highest power of in the denominator and proceed from there. Find the its in Eercises : q A : -q : q : -q + 5 r: q : q A : q. 5> - > + 7. : q 8>5 + + cos u u: -q ƒsd = > + - : -q : q - : -q 8 - s5> d u r + sin r r sin r > -

19 .6 Limits Involving Infinit; Asmptotes of Graphs : q + 5 Infinite Limits Find the its in Eercises : :7 s - 7d 5. a. b. : + > : - > 6. a. b. : + >5 : - > Find the its in Eercises tan 5. :sp>d - 5. s + csc ud 5. - Find the its in Eercises as a. : + b. : - c. : - + d. : as : q : - : -8 + : u: >5 a. : + b. : - c. : - + d. : - - a - b as : -q : - 5 : + - : : a. : + b. : - c. : d. : as a. : - + b. : - - c. : + d. : as a. : + b. : + c. : - d. : + : -q + : > :s-p>d + sec s - cot ud u: e. What, if anthing, can be said about the it as :? as a. : + b. : - + c. : - d. : s + d e. What, if anthing, can be said about the it as :? Find the its in Eercises a - b as > t a. t : + b. t : a + 7b as >5 t a. t : + b. t : - a + b as > > s - d a. : + b. : - c. : + d. : - a - b as > > s - d a. : + b. : - c. : + d. : - Graphing Simple Rational Functions Graph the rational functions in Eercises Include the graphs and equations of the asmptotes and dominant terms. 6. = 6. = = = = + Inventing Graphs and Functions In Eercises 69 7, sketch the graph of a function = ƒsd that satisfies the given conditions. No formulas are required just label the coordinate aes and sketch an appropriate graph. (The answers are not unique, so our graphs ma not be eactl like those in the answer section.) 69. ƒsd =, ƒsd =, ƒs -d = -, ƒsd = -, and ƒsd = : -q : q 7. ƒsd =, ƒsd =, ƒsd =, and In Eercises 7 76, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. An function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) : ƒsd =, ƒsd =, ƒsd = ƒsd = q, - + : ;q : : - ƒsd = -q, and ƒsd = -q + - : ƒsd =, ƒs -d =, : : ;q : ;q 75. hsd = -, hsd =, hsd = -, and - : -q : ;q - ƒsd = - ƒsd = -q, and ƒsd = - : -q ƒsd =, ƒsd = q, and ƒsd = q - : : + gsd =, gsd = -q, and gsd = q - : : + : q : - : q : = - - ƒsd =, ƒsd = q, + : : + hsd = : ksd =, ksd = q, and k sd = -q - + : ;q : :

20 6 Chapter : Limits and Continuit 77. Suppose that ƒ() and g() are polnomials in and that :q sƒsd>gsdd =. Can ou conclude anthing about :-q sƒsd>gsdd? Give reasons for our answer. 78. Suppose that ƒ() and g() are polnomials in. Can the graph of ƒ()>g() have an asmptote if g() is never zero? Give reasons for our answer. 79. How man horizontal asmptotes can the graph of a given rational function have? Give reasons for our answer. Finding Limits of Differences when Find the its in Eercises A B : q 8. A : q B A : -q + + B A + : -q + - B A : q B A : q B A : q B Using the Formal Definitions Use the formal definitions of its as : ; q to establish the its in Eercises 87 and If ƒ has the constant value ƒsd = k, then 88. If ƒ has the constant value ƒsd = k, then Use formal definitions to prove the it statements in Eercises : = -q : s - d = -q 9. : ; ˆ : : -5 : q : -q = q ƒ ƒ s + 5d = q 9. Here is the definition of infinite right-hand it. ƒsd = k. ƒsd = k. We sa that ƒ() approaches infinit as approaches from the right, and write ƒsd = q, + : if, for ever positive real number B, there eists a corresponding number d 7 such that for all d Q ƒsd 7 B. T T Modif the definition to cover the following cases. a. b. c. Use the formal definitions from Eercise 9 to prove the it statements in Eercises : = q = -q 98. ƒsd = q - : ƒsd = -q + : ƒsd = -q - : : - : - - = q Oblique Asmptotes Graph the rational functions in Eercises 99. Include the graphs and equations of the asmptotes = + = - -. = = -. Additional Graphing Eercises Graph the curves in Eercises 5 8. Eplain the relationship between the curve s formula and what ou see. 5. = 6. = p = sin a + b > Graph the functions in Eercises 9 and. Then answer the following questions. a. How does the graph behave as : +? b. How does the graph behave as : ; q? c. How does the graph behave near = and = -? Give reasons for our answers. : = -q - : + - = q = - + = = > a = a - > b - b Chapter Questions to Guide Your Review. What is the average rate of change of the function g(t) over the interval from t = a to t = b? How is it related to a secant line?. What it must be calculated to find the rate of change of a function g(t) at t = t?. Give an informal or intuitive definition of the it ƒsd = L. : Wh is the definition informal? Give eamples.