1 DL3. Infinite Limits and Limits at Infinity
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1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 78 Mark Sparks 01 Infinite Limits and Limits at Infinity In our graphical analysis of its, we have already seen both an infinite it and a it at infinity. Let s consider the equations and the graphs of the two functions below to find the its that follow f 1 g Infinite Limits Limits at Infinity f f f f 1 g g 1 g g 1 DL3
2 We have already seen how to find infinite its by marrying a numerical and analytical approach. For the function, f and g, whose graphs appear on the previous page, find the infinite its below Graphically, an infinite it will always yield a. In pre-calculus, we discovered through observation that such a graphical property eisted when a factor in the equation would not. From this point forward, this is NOT a viable justification for the eistence of a. Justification of the Eistence of a Vertical Asymptote Using Limits For the function below, find any vertical asymptotes that eist. Justify your answers using a its. h Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 79 Mark Sparks 01
3 Now, we must develop an analytical procedure by which we can find its at infinity. Basically, a it at infinity describes the end behavior of a function. We have spent a great amount of time talking about end behavior of functions. Find each of the following its at infinity. Give an eplanation of your reasoning for each The third eample, will provide us a basis for developing our analytical process by which we 3 can find its at infinity for all types of rational functions. Before we do that, investigate the two functions below both graphically and numerically. 3 f g 1 What does each of these functions have in common algebraically and what do they have in common graphically? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 80 Mark Sparks 01
4 In pre-calculus, we learned three rules for determining the eistence of horizontal asymptotes of rational functions. When a rational function had a horizontal asymptote, the end behavior was always such that as or, then the graph of f the horizontal asymptote. We learned three rules for determining the horizontal asymptote, if one eisted, for rational functions. We are about to use the idea of a it and calculus to find out why those rules are such as they are. For each function below, divide every term in both the numerator and the denominator by the highest power of that appears in the denominator. Then, evaluate the indicated it. Does the result of each it make sense based on the graph that is pictured? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 81 Mark Sparks 01
5 Let s see what would happen if our it at infinity approached Based on what we have just seen and what we know graphically about the functions above, would does approaching make a difference in the result of our its? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 8 Mark Sparks 01
6 Graphically, a it at infinity can yield a. In pre-calculus, we discovered through observation that such a graphical property eisted by comparing the. From this point forward, this is NOT a viable justification for the eistence of a. Justification of the Eistence of a Horizontal Asymptote Using Limits For the function below, find the horizontal asymptote if it eists. Justify your answers using a its. 5 h 3 3 The algebraic analysis described above to evaluate a it at infinity can be used to find its at infinity for any type of rational function, even f, whose graph is pictured to the right. 1 What is the one thing that you notice is different about the graph of this rational function versus the others that we have investigated in the past? Use the graph to find each of the following its. 1 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 83 Mark Sparks 01
7 Perform the same algebraic analysis that we did earlier to find the its at infinity. The only problem that we will encounter is what to do when. 1 1 Look at the graph on the previous page to confirm these results. Then, find the its below Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 84 Mark Sparks 01
8 AP Calculus Multiple Choice Practice Graphing Calculator NOT Permitted 0 minutes 1.. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 85 Mark Sparks 01
9 3. 4. If f = 3 +, then h 0 f h h f is A 6 + B 6 C 0 D noneistent E 5. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 86 Mark Sparks 01
10 sin, 4 g,, For the function above, which of the following would be the reasons why the function, g, is not continuous at = 3? I. g3 is undefined. II. g does not eist. III. g g A III only B II only C I and II only D I only E II and III only 7. 3 is A 0 B C D E ½ 8. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 87 Mark Sparks 01
11 Consider the function a. Find h AP Calculus Free Response Practice #1 Calculator Permitted sin h to answer the following questions. 1. Show your analysis. b. Identify the vertical asymptotes, if any eist, of h and justify the eistence by writing a it. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 88 Mark Sparks 01
12 c. Identify the horizontal asymptotes, if any eist, of h and justify the eistence by writing a it. d. Eplain why the Intermediate Value Theorem guarantees a value of c on the interval [1.5,.5] such that hc = 4. Then, find c. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 89 Mark Sparks 01
13 AP Calculus Free Response Practice # Calculator NOT Permitted a 3, f 3, b 5, 3 3 Graph of g Equation of f Pictured above is the graph of a function g and the equation of a piece-wise defined function f. Answer the following questions. a. Find g f cos 1. Show your work applying the properties of its. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 90 Mark Sparks 01
14 b. On its domain, what is one value of at which g is discontinuous? Use the three part definition of continuity to eplain why g is discontinuous at this value. c. For what values of a and b, if they eist, would the function f be continuous everywhere? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 91 Mark Sparks 01
15 AP Calculus Etra Practice on Limits and Multiple Choice Practice For questions 1 5, refer to the graph of f to the right. Find the value of each indicated it. If a it does not eist, give a reason. 1. f 3 f f cos 1 3. f 4 4. f 5. f 3 For questions 6 11, find the value of each it analytically. If a it does not eist, state why tan 0 sec ln Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 9 Mark Sparks 01
16 For question 1 16, use the equation g below and the graph of the function f. Graph of f 3 3, g cos, a, 1. Is g is continuous at =. [Base your response on the three part definition of continuity.] ` 13. For what values of a is g continuous at =? 14. For what values of b is the function f discontinuous? At which of these values does f b eist? Eplain your reasoning. 15. Find [ g f ]. 16. Which of the following its does not eist? Give a reason for your answers. f f f Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 93 Mark Sparks 01
17 17. Find the values of k and m so that the function below is continuous on the interval,. k3, f 3, 3 m, A. B. C. 0 D. 7 4 E A. B. C. 0 D. E A. 4 B. 0 C. 1 D. 3 E. 1. The function 3, G 5, 3 7, is not continuous at = because A. G is not defined B. G does not eist C. G G D. Only reasons B and C E. All of the above reasons A. B. C. 1 D. 7 E. 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 94 Mark Sparks 01
18 A. 1 B. 0 C. D. E. Does Not Eist f h f 4. If f 3 5, then find. 0 h A. 3 5 B. 6 5 C. 6 D. 0 E. Does not eist h 5 5. A. 5 B. 5 C. 0 D. E. 6. The function f 3 has a vertical asymptote at = 5 because 4 5 A. 5 f C. 5 f B. f 5 D. f 5 E. f does not have a vertical asymptote at = 5 7. Consider the function 3 5, 3 H. Which of the following statements is/are true?, 3 I. H 4. II. H eists. III. H is continuous at = A. I only B. II only C. I and II only D. I, II and III E. None of these statements is true Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 95 Mark Sparks 01
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