Practice Set 26 Limits and Continuity II

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1 Practice Set 6 Limits and Continuity II No Calculator Objectives Determine where a rational function is discontinuous and the type of discontinuity. Find the equation of the vertical asymptote of a rational function s graph. Find the equation of the horizontal asymptote of a rational function s graph. Find the coordinates of a hole in a rational function s graph. Notes Rational Function A quotient of polynomials such that the denominator has a degree of at least 1. End Behavior of Rational Functions If the degree of the numerator is greater than the degree of the denominator, the end behavior will be or. If the degree of the numerator is equal to the degree of the denominator, the end behavior will be a horizontal asymptote of F() = L. If the degree of the numerators is less than the degree of the denominator, the end behavior will be a horizontal asymptote of f() = 0. Evaluating Infinite Limits of Rational Functions Select the term with the largest degree in the numerator. Select the term with the largest degree in the denominator. Simplify the epression and evaluate the limit. 1 Definition: lim 0 1. (ACT/SAT) Harry took a loan from the bank. Each month, he pays a fied amount of money back to the bank. The equation D 00t 9000 shows the remaining balance of the loan, D, measured in dollars, after t months. How much does Harry pay back to the bank each month, in dollars?. (ACT/SAT) Andrei has a glass tank. First, he wants to put some marbles in it, all of the same volume. Then, he wants to fill the tank with water until it s completely full. The equation W 0.05n describes the volume of water, W, measured in liters, that Andrei should use when there are n marbles. What is the volume of the glass tank, in liters? For problems -, rewrite each epression in the form terms o. b a e, where a is a coefficient and b is an eponent in. (9) 1 e 5e. (9) e 1e 51 For problems 5-6, given the composition of functions below, identify the outer function and the inner function (1) f g 5 6. (1) g e 5e 1 Inner Function: Outer Function: Inner Function: Outer Function: Rational Functions Investigations 19

2 For problems 7-8, use the piecewise function below to evaluate each of the following: (0) f8 8. (0) f 6 For problems 9-16, determine where f() is discontinuous. Find the type of each discontinuity and the implications on the graph. 9. f f f f f f 5 10 Rational Functions Investigations

3 For problems 17-8, evaluate each limit. 17. lim lim lim 0. 1 lim 0 1. lim lim lim 9. lim lim 1 6. lim lim 8. lim? 16 I. = 1 II. = III. = 9. Multiple choice Which of the following represent the vertical asymptote(s) of A. I only B. II only C. III only D. II and III only 0. Multiple choice Which of the following is the horizontal asymptote of 1? 1 A. 0 B. C. D Multiple choice Which of the following is a point discontinuity of 9? 5 A., 1 B. 1,0 C. 9, 5 D. 1,0. Multiple choice Which of the following has a horizontal asymptote of? I. f 1 1 II. f 9 1 III. 1 1 A. I only B. II only C. III only D. I and III only Rational Functions Investigations 11

4 . Multiple choice Which of the following will have a horizontal asymptote? I. f II. f 9 III. 8 7 A. I and II only B. II and III only C. I and III only D. I, II, III. Select all that apply Which of the following functions has at least one infinite discontinuity? A. f B. f 16 C. 1 D. f 5. Select all that apply Which of the following functions has a point discontinuity? 9 A. B. f 1 C. f D. f Select all that apply The graph of which of the functions below will have a hole? A. f 6 B. C. 5 5 D. f 1 7. Select all that apply The graph of which of the functions below will have a horizontal asymptote? A B. f C. f D. f 8. Select all that apply The graph of which of the functions below will have a vertical asymptote? A. f 5 5 B. f 1 C. f 8 1 D. 1 Rational Functions Investigations

5 Practice Set 7 Graphing Rational Functions No Calculator Objectives Graph a rational function given: o intercepts, asymptotes, and holes (if any). o the equation of a rational function Notes Horizontal intercept(s): Set 0 and solve for. Vertical intercept(s): The value of f0 Vertical asymptote/infinite discontinuity: lim Horizontal asymptote: lim a a is undefined,, or. horizontal asymptote at Hole(s): fa does not eist, but lim f lim a a 1. (ACT/SAT) Quinn returned home one summer s day to find it etremely hot. He turned the air conditioner on, and the room s temperature began decreasing at a constant rate. The equation R 0.7t gives the room s temperature, R, in degrees Celsius, t minutes after Quinn turned on the air conditioner. What was the room s temperature, in degrees Celsius, when Quinn returned home? a. (ACT/SAT) A big cruise ship drops anchor off the Caribbean island of Antigua. The heavy anchor drops into the water at a constant rate. The equation E.t 75 gives the anchor s elevation, E, measured in meters relative to the water s surface, after t seconds. How many meters does the anchor drop every 5 seconds? For problems -, rewrite each epression in the form terms o. b a e, where a is a coefficient and b is an eponent in. (9) e e 5. (9) 6e 9e 9 6 For problems 5-6, given the composition of functions below, identify the outer function and the inner function. 5. (1) f g 5 6. (1) g 6 5 Inner Function: Outer Function: Inner Function: Outer Function: Rational Functions Investigations 1

6 For problems 7-8, use the piecewise function below to evaluate each of the following: (0) f 8. (0) f6 For problems 9-19, provide the information where needed and graph the function. 9. f 5 horizontal intercept(s): none vertical intercept(s): 5 infinite discontinuity vertical asymptote at = 5 lim lim 5 5 horizontal asymptote at f() = f 1 6 horizontal intercept(s): none vertical intercept(s): 1 6 infinite discontinuity vertical asymptote at = 6 lim lim 6 6 horizontal asymptote at f() = 0 1 Rational Functions Investigations

7 1 11. horizontal intercept(s): vertical intercept(s): infinite discontinuity vertical asymptote at = lim lim horizontal asymptote at f() = 1. f 6 horizontal intercept(s): 1 1 vertical intercept(s): infinite discontinuity vertical asymptote at = lim lim horizontal asymptote at f() = 1 point discontinuity hole at, 5 1. g 9 horizontal intercept(s): none 1 vertical intercept(s): infinite discontinuity vertical asymptote at = lim g lim g horizontal asymptote at g() = 0 point discontinuity 1 hole at, 6 Rational Functions Investigations 15

8 1. f 5 horizontal intercept(s) horizontal asymptote(s) 15. f 1 9 horizontal intercept(s) horizontal asymptote(s) 16. f horizontal intercept(s) horizontal asymptote(s) 16 Rational Functions Investigations

9 17. horizontal intercept(s): omit horizontal asymptote(s) 18. f horizontal intercept(s) horizontal asymptote(s) 19. f horizontal intercept(s) horizontal asymptote(s) Rational Functions Investigations 17

10 Practice Set 8 The Slant Asymptote No Calculator Objectives Identify rational functions whose graph contains a slant asymptote. Find the equation of the slant asymptote when it eists. Graph a rational function with a slant asymptote. Notes In a rational function, if the degree of the numerator is eactly one more than the degree of the denominator, the graph of the function will have a slant asymptote. The equation of the slant asymptote is the result of numerator divided by denominator (ignoring the remainder). 5 f has a slant asymptote: y A function whose graph has a slant asymptote will not have a horizontal asymptote. 1. (ACT/SAT) A lake near the Arctic Circle is covered by a thick sheet of ice during the cold winter months. When Spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a constant rate. The equation S 0.5t gives the ice sheet s thickness, S, measured in meters, t weeks after it begins to melt. By how many meters does the sheet s thickness decrease every 6 weeks?. (ACT/SAT) Rachel is driving a race car at a constant speed on a closed course. The formula D 8t 0 describes the remaining distance, D, measured in meters, that Rachel has to travel after t seconds. What is Rachel s speed in meters per second? For problems -, rewrite each epression in the form terms o. b a e, where a is a coefficient and b is an eponent in. (9) 1 7 5e e. (9) 1e 7e 67 For problems 5-6, given the composition of functions below, identify the outer function and the inner function (1) f g 5 6. (1) g 1 1 Inner Function: Outer Function: Inner Function: Outer Function: 18 Rational Functions Investigations

11 For problems 7-8, use the piecewise function below to evaluate each of the following: (0) f 8. (0) f5 For problems 9-1, complete each division using the area model Rational Functions Investigations 19

12 1. Select all that apply The graph of which of the functions below will have a slant asymptote? A. 5 B. f 6 C. f 6 D. f Select all that apply The graph of which of the functions below will have a slant asymptote? 9 A. B. f 1 C. f D. f Select all that apply The graph of which of the functions below will have a slant asymptote? A. f 1 B. f C. f 7 1 D. f Multiple choice Which of the following is the slant asymptote of f 1? A. y B. y C. y D. y 17. Multiple choice Which of the following is the slant asymptote of f 1? A. y B. y C. y D. y 18. Multiple choice Which of the following is the slant asymptote of f 1? A. y 1 B. y 1 C. y 1 D. y Rational Functions Investigations

13 For problems 19-, complete the information and graph for each function below. 19. f 6 1 horizontal intercept(s):, vertical intercept(s): 6 infinite discontinuity vertical asymptote at = 1 lim lim 1 1 horizontal/slant asymptote: y point discontinuity no holes 0. f 1 horizontal intercept(s): 1, vertical intercept(s): 1 1 infinite discontinuity vertical asymptote at = 1 lim lim 1 1 horizontal/slant asymptote: y 5 point discontinuity hole:,1 1. f 1 1 horizontal intercept(s): 1 vertical intercept(s) 1 infinite discontinuity vertical asymptote at = 1 lim lim 1 1 horizontal/slant asymptote: 1 5 y point discontinuity no holes Rational Functions Investigations 151

14 . f 1 horizontal intercept(s) horizontal or slant asymptote(s). f 1 horizontal intercept(s) horizontal or slant asymptote(s). f 1 horizontal intercept(s) horizontal or slant asymptote(s) 15 Rational Functions Investigations

15 Practice Set 9 Assessment 7 Review - 50 Points 1. (ACT/SAT) A technician is transferring files from a mainframe onto a flash drive at a constant rate. The equation S 5t 5 gives the number of megabytes, S, on the flash drive after t seconds. How many additional megabytes are transferred onto the flash drive every 10 seconds?. (ACT/SAT) The equation F 1.8C describes the temperature, F, in degrees Fahrenheit, when the temperature is C degrees Celsius. If the temperature rises 10 degrees Celsius, by how much does the temperature rise in degrees Fahrenheit? Section 9 Eponential Epressions ( pts) (No Calculator) Compare values of linear epressions, epressions with eponents, and eponential epressions. Change the base of an eponential epression Combine eponential epressions For problems -, rewrite each epression in the form terms o. b a e, where a is a coefficient and b is an eponent in. (9) e e 1. (9) 1e 7e 67 Section 1 Composition of Functions ( pts) (No Calculator) Compute the value of the composition of two functions (linear, quadratic, and/or eponential). Create a new function as the composition of two other functions (linear, quadratic, polynomial, and/or eponential). Given the composition of two functions, identify the inner function and outer function. For problems 5-6, given the composition of functions below, identify the outer function and the inner function. 5. (1) f g (1) g 5 Inner Function: Outer Function: Inner Function: Outer Function: Rational Functions Investigations 15

16 Section 0 Understanding the Piecewise Function ( pts) (No Calculator) Given a value o, identify which piece of a piecewise function should be used to evaluate. Evaluate a piecewise function given a graph. Evaluate a piecewise function given an equation. For problems 7-8, use the piecewise function below to evaluate each of the following: (0) f6 8. (0) f 5 Section 6 Limits and Continuity II (15 pts) (No Calculator) Determine where a rational function is discontinuous and the type of discontinuity. Find the equation of the vertical asymptote of a rational function s graph. Find the equation of the horizontal asymptote of a rational function s graph. Find the coordinates of a hole in a rational function s graph. Section 7 Graphing Rational Functions (18 pts) (No Calculator) Graph a rational function given: o intercepts, asymptotes, and holes (if any). o the equation of a rational function Section 8 The Slant Asymptote (9 pts) (/No Calculator) Identify a rational function whose graph will contain a slant asymptote. Find the equation of the slant asymptote of a rational function when it eists. Graph a rational function which contains a slant asymptote. For problems 9-1, determine where f() is discontinuous, if anywhere. Find the type of each discontinuity and the implications on the graph. 9. f 10. f 11. f Rational Functions Investigations

17 For problems 1-16, evaluate each limit. 1. lim 1 1. lim lim lim 17. Select all that apply Which of the following functions has a point discontinuity? A. f B. f 16 C. 1 D. f 18. Multiple choice Which of the following functions has point discontinuity? I. f II. f 8 III. A. I and II only B. II and III only C. I and III only D. I, II, III 19. Select all that apply Which of the following functions has infinite discontinuity? A. f 6 B. C. 5 5 D. f 1 0. Multiple choice Which of the following functions has infinite discontinuity? I. f 1 II. f 1 III. 11 A. I and II only B. I and III only C. II and III only D. I, II, III 1. Select all that apply The graph of which of the functions below will have a horizontal asymptote? A. f B. f 16 C. 1 D. f 1 Rational Functions Investigations 155

18 . Multiple choice The graph of which of the functions below will have a horizontal asymptote? I. f II. f 9 III. 8 7 A. I and II only B. II and III only C. I and III only D. I, II, III. Select all that apply The graph of which of the functions below will have a vertical asymptote? A B. f C. f 9 D. f. Multiple choice The graph of which of the functions below will have a vertical asymptote? I. f II. f 8 III. A. I only B. I and II only C. I and III only D. II and III only 5. Select all that apply The graph of which of the functions below will have a hole? A. f B. f 16 C. 1 D. f 6. Multiple choice The graph of which of the functions below will have a hole? I. f 5 6 II. III A. I only B. II only C. III only D. I and II only 7. Select all that apply The graph of which of the functions below will have a slant asymptote? A. 5 B. f 6 C. f 6 D. f Multiple choice The graph of which of the functions below will have a slant asymptote? I. f 1 II. f 1 III. 11 A. I only B. I and II only C. I and III only D. II and III only 156 Rational Functions Investigations

19 For problems 9-, provide the information where needed and graph the function horizontal intercept(s): none vertical intercept(s): 5 infinite discontinuity vertical asymptote at = 5 lim lim 5 5 horizontal/slant asymptote: f() = 0 point discontinuity hole at 6, f horizontal intercept(s): 0 vertical intercept(s): 0 infinite discontinuity vertical asymptote at = lim lim horizontal asymptote at f() = 1 point discontinuity 1 hole at, 1. f 1 horizontal intercept(s): 0 vertical intercept(s): 0 infinite discontinuity vertical asymptote at = 1 lim g lim g 1 1 slant asymptote: y 1 Rational Functions Investigations 157

20 1. horizontal intercept(s) horizontal/slant asymptote. f 1 horizontal intercept(s) horizontal/slant asymptote. f 6 1 horizontal intercept(s) horizontal/slant asymptote 158 Rational Functions Investigations

21 Answers to Selected Eercises Practice Set 6 Limits and Continuity II P e. e 7 5. Inner: g 5 ; Outer: 5 6. Inner: e ; Outer: g infinite discontinuity/vertical asymptote at = 10. point 1 discontinuity/hole at, 6 ; infinite discontinuity/vertical asymptote at = 11. point discontinuity/hole at,1 ; infinite discontinuity/vertical asymptote at = 1. infinite discontinuity/vertical asymptote at = 5 and = infinite discontinuity/vertical asymptote at = 1 and = 1. point discontinuity/hole at, 1, ; infinite 15. infinite discontinuity/vertical asymptote at = and = 16. point discontinuity/hole at discontinuity/vertical asymptote at = B 0. D 1. A. D. C. A, C 5. A, B, C 6. B, C 7. A, B, D 8. B, C, D Practice Set 7 Graphing Rational Functions P e 5. e 5. Inner: g ; Outer: Inner: 6 ; Outer: g UND H.I. none V.A. = -5 V.I. 5 H.A. f() = 0 Hole(s) none 15. H.I. V.A. = 16. H.I. none V.A. = -1 V.I. H.A. f() = 1 V.I. 1 H.A. f() = 0 7 Hole(s), H.I. omit V.A. = 18. V.I. 7 H.A. f() = 17 Hole(s) 1, H.I. V.A. = H.I. 1 1 V.I. 1 Hole(s), V.A. = - H.A. f() = 9 Hole(s), V.I. 1 H.A. f() = 1 1 Hole(s): 5, Rational Functions Investigations 159

22 Practice Set 8 The Slant Asymptote P e 10. e 5. Inner: g 1; Outer: 5 6. Inner: 1; Outer: g B, D 1. B 15. A, B, D 16. C 17. B 18. D H.I. 1, 1 V.A. = 0 V.I. none S.A. y = Hole(s) none 5 5. H.I. 0, V.A. = 1. H.I. 0 V.A. = 1, = 1 V.I. 0 S.A. y = 5 Hole(s) none V.I. 0 S.A. y = Hole(s) none Practice Set 9 Assessment 7 Review P e 10. e 5. Inner: g 6 ; Outer: 6. Inner: 5 ; Outer: g infinite discontinuity/vertical asymptote at 5 = 10. infinite discontinuity/vertical asymptote at = ; point discontinuity/hole at, infinite discontinuity/vertical asymptote at = 7 and = 5 1. continuous A, B, D 18. D 19. A, B, C, D 0. B 1. A, C. C. A, B, D. C 5. A, B 6. D 7. B, D 8. D H.I. 1 V.A. = V.I. 1 H.A. f() = 1 Hole(s):,. H.I. V.A. = 1. V.I. H.A. f() = 5 Hole(s): 1, H.I., V.A. = 1 V.I. 6 S.A. y 1 Hole(s): none 160 Rational Functions Investigations