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# Lesson 5: Limits Involving Infinity

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### Lesson 5: Limits Involving Infinity

1. 1. Section 1.6 Limits involving Inﬁnity V63.0121.002.2010Su, Calculus I New York University May 20, 2010 Announcements Oﬃce Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here) Quiz 1 Thursday on 1.1–1.4
2. 2. Announcements Oﬃce Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here) Quiz 1 Thursday on 1.1–1.4 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 2 / 37
3. 3. Objectives “Intuit” limits involving inﬁnity by eyeballing the expression. Show limits involving inﬁnity by algebraic manipulation and conceptual argument. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 3 / 37
4. 4. Recall the deﬁnition of limit Deﬁnition We write lim f (x) = L x→a and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x to be suﬃciently close to a (on either side of a) but not equal to a. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 4 / 37
5. 5. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y L? x No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
6. 6. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y L? x No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
7. 7. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y L? x No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
8. 8. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y L? x No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
9. 9. Outline Inﬁnite Limits Vertical Asymptotes Inﬁnite Limits we Know Limit “Laws” with Inﬁnite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Rationalizing to get a limit V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 6 / 37
10. 10. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. x V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
11. 11. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. “Large” takes the place of x “close to L”. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
12. 12. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. “Large” takes the place of x “close to L”. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
13. 13. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. “Large” takes the place of x “close to L”. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
14. 14. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. “Large” takes the place of x “close to L”. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
15. 15. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. “Large” takes the place of x “close to L”. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
16. 16. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. “Large” takes the place of x “close to L”. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
17. 17. Inﬁnite Limits Deﬁnition The notation y lim f (x) = ∞ x→a means that values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. “Large” takes the place of x “close to L”. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
18. 18. Negative Inﬁnity Deﬁnition The notation lim f (x) = −∞ x→a means that the values of f (x) can be made arbitrarily large negative (as large as we please) by taking x suﬃciently close to a but not equal to a. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 8 / 37
19. 19. Negative Inﬁnity Deﬁnition The notation lim f (x) = −∞ x→a means that the values of f (x) can be made arbitrarily large negative (as large as we please) by taking x suﬃciently close to a but not equal to a. We call a number large or small based on its absolute value. So −1, 000, 000 is a large (negative) number. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 8 / 37
20. 20. Vertical Asymptotes Deﬁnition The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following is true: lim f (x) = ∞ lim f (x) = −∞ x→a x→a lim+ f (x) = ∞ lim f (x) = −∞ x→a x→a+ lim f (x) = ∞ lim f (x) = −∞ x→a− x→a− V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 9 / 37
21. 21. Inﬁnite Limits we Know y 1 lim+ =∞ x→0 x x V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 10 / 37
22. 22. Inﬁnite Limits we Know y 1 lim+ =∞ x→0 x 1 lim = −∞ − x x→0 x V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 10 / 37
23. 23. Inﬁnite Limits we Know y 1 lim+ =∞ x→0 x 1 lim = −∞ − x x→0 x 1 lim 2 = ∞ x→0 x V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 10 / 37
24. 24. Finding limits at trouble spots Example Let x2 + 2 f (x) = x 2 − 3x + 2 Find lim f (x) and lim+ f (x) for each a at which f is not continuous. x→a− x→a V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 11 / 37
25. 25. Finding limits at trouble spots Example Let x2 + 2 f (x) = x 2 − 3x + 2 Find lim f (x) and lim+ f (x) for each a at which f is not continuous. x→a− x→a Solution The denominator factors as (x − 1)(x − 2). We can record the signs of the factors on the number line. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 11 / 37
26. 26. Use the number line − 0 + (x − 1) 1 So
27. 27. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 So
28. 28. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) So
29. 29. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) f (x) 1 2 So
30. 30. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + f (x) 1 2 So
31. 31. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ f (x) 1 2 So lim f (x) = +∞ x→1−
32. 32. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ −∞ f (x) 1 2 So lim f (x) = +∞ x→1− lim f (x) = −∞ x→1+
33. 33. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ −∞ − f (x) 1 2 So lim f (x) = +∞ x→1− lim f (x) = −∞ x→1+
34. 34. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ −∞ − −∞ f (x) 1 2 So lim f (x) = +∞ lim f (x) = −∞ x→1− x→2− lim f (x) = −∞ x→1+
35. 35. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ −∞ − −∞ +∞ f (x) 1 2 So lim f (x) = +∞ lim f (x) = −∞ x→1− x→2− lim f (x) = −∞ lim f (x) = +∞ x→1+ x→2+
36. 36. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ −∞ − −∞ +∞ f (x) 1 2 So lim f (x) = +∞ lim f (x) = −∞ x→1− x→2− lim f (x) = −∞ lim f (x) = +∞ x→1+ x→2+
37. 37. Use the number line − 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x 2 + 2) + +∞ −∞ − −∞ +∞ + f (x) 1 2 So lim f (x) = +∞ lim f (x) = −∞ x→1− x→2− lim f (x) = −∞ lim f (x) = +∞ x→1+ x→2+ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 12 / 37
38. 38. In English, now To explain the limit, you can say: “As x → 1− , the numerator approaches 3, and the denominator approaches 0 while remaining positive. So the limit is +∞.” V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 13 / 37
39. 39. The graph so far y x −1 1 2 3 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
40. 40. The graph so far y x −1 1 2 3 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
41. 41. The graph so far y x −1 1 2 3 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
42. 42. The graph so far y x −1 1 2 3 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
43. 43. The graph so far y x −1 1 2 3 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
44. 44. Limit Laws (?) with inﬁnite limits If lim f (x) = ∞ and lim g (x) = ∞, then lim (f (x) + g (x)) = ∞. x→a x→a x→a That is, ∞+∞=∞ If lim f (x) = −∞ and lim g (x) = −∞, then x→a x→a lim (f (x) + g (x)) = −∞. That is, x→a −∞ − ∞ = −∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 15 / 37
45. 45. Rules of Thumb with inﬁnite limits If lim f (x) = ∞ and lim g (x) = ∞, then lim (f (x) + g (x)) = ∞. x→a x→a x→a That is, ∞+∞=∞ If lim f (x) = −∞ and lim g (x) = −∞, then x→a x→a lim (f (x) + g (x)) = −∞. That is, x→a −∞ − ∞ = −∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 15 / 37
46. 46. Rules of Thumb with inﬁnite limits If lim f (x) = L and lim g (x) = ±∞, then lim (f (x) + g (x)) = ±∞. x→a x→a x→a That is, L+∞=∞ L − ∞ = −∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 16 / 37
47. 47. Rules of Thumb with inﬁnite limits Kids, don’t try this at home! The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite limit is not 0. ∞ if L > 0 L·∞= −∞ if L < 0. −∞ if L > 0 L · (−∞) = ∞ if L < 0. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 17 / 37
48. 48. Multiplying inﬁnite limits Kids, don’t try this at home! The product of two inﬁnite limits is inﬁnite. ∞·∞=∞ ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 18 / 37
49. 49. Dividing by Inﬁnity Kids, don’t try this at home! The quotient of a ﬁnite limit by an inﬁnite limit is zero: L =0 ∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 19 / 37
50. 50. Dividing by zero is still not allowed 1 =∞ 0 There are examples of such limit forms where the limit is ∞, −∞, undecided between the two, or truly neither. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 20 / 37
51. 51. Indeterminate Limit forms L Limits of the form are indeterminate. There is no rule for evaluating 0 such a form; the limit must be examined more closely. Consider these: 1 −1 lim =∞ lim = −∞ x→0 x2 x→0 x2 1 1 lim+ = ∞ lim = −∞ x→0 x x→0− x 1 L Worst, lim is of the form , but the limit does not exist, even x sin(1/x) x→0 0 in the left- or right-hand sense. There are inﬁnitely many vertical asymptotes arbitrarily close to 0! V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 21 / 37
52. 52. Indeterminate Limit forms Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate. Example 1 The limit lim+ sin x · is of the form 0 · ∞, but the answer is 1. x→0 x 1 The limit lim+ sin2 x · is of the form 0 · ∞, but the answer is 0. x→0 x 1 The limit lim+ sin x · 2 is of the form 0 · ∞, but the answer is ∞. x→0 x Limits of indeterminate forms may or may not “exist.” It will depend on the context. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 22 / 37
53. 53. Indeterminate forms are like Tug Of War Which side wins depends on which side is stronger. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 23 / 37
54. 54. Outline Inﬁnite Limits Vertical Asymptotes Inﬁnite Limits we Know Limit “Laws” with Inﬁnite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Rationalizing to get a limit V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 24 / 37
55. 55. Limits at inﬁnity Deﬁnition Let f be a function deﬁned on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x suﬃciently large. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 25 / 37
56. 56. Limits at inﬁnity Deﬁnition Let f be a function deﬁned on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x suﬃciently large. Deﬁnition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x→∞ x→−∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 25 / 37
57. 57. Limits at inﬁnity Deﬁnition Let f be a function deﬁned on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x suﬃciently large. Deﬁnition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x→∞ x→−∞ y = L is a horizontal line! V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 25 / 37
58. 58. Basic limits at inﬁnity Theorem Let n be a positive integer. Then 1 lim =0 x→∞ x n 1 lim =0 x→−∞ x n V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 26 / 37
59. 59. Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 27 / 37
60. 60. Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 27 / 37
61. 61. Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim 3 + 5x 2 + 7 = lim x→∞ 4x x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 28 / 37
62. 62. Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim 3 + 5x 2 + 7 = lim x→∞ 4x x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 Upshot When ﬁnding limits of algebraic expressions at inﬁnity, look at the highest degree terms. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 28 / 37
63. 63. Another Example Example x Find lim x→∞ x2 + 1 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 29 / 37
64. 64. Another Example Example x Find lim x→∞ x2 + 1 Answer The limit is 0. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 29 / 37
65. 65. Solution Again, factor out the largest power of x from the numerator and denominator. We have x x(1) 1 1 = 2 = · x2 +1 x (1 + 1/x 2) x 1 + 1/x 2 x 1 1 1 1 lim 2 = lim = lim · lim x→∞ x + 1 x→∞ x 1 + 1/x 2 x→∞ x x→∞ 1 + 1/x 2 1 =0· = 0. 1+0 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 30 / 37
66. 66. Another Example Example x Find lim x→∞ x2 + 1 Answer The limit is 0. y x V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 31 / 37
67. 67. Another Example Example x Find lim x→∞ x2 + 1 Answer The limit is 0. y x Notice that the graph does cross the asymptote, which contradicts one of the heuristic deﬁnitions of asymptote. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 31 / 37
68. 68. Solution Again, factor out the largest power of x from the numerator and denominator. We have x x(1) 1 1 = 2 = · x2 +1 x (1 + 1/x 2) x 1 + 1/x 2 x 1 1 1 1 lim 2 = lim = lim · lim x→∞ x + 1 x→∞ x 1 + 1/x 2 x→∞ x x→∞ 1 + 1/x 2 1 =0· = 0. 1+0 Remark Had the higher power been in the numerator, the limit would have been ∞. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 32 / 37
69. 69. Another Example Example Find √ 3x 4 + 7 lim x→∞ x2 + 3 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 33 / 37
70. 70. Another Example Example Find √ √ 3x 4 + 7 ∼ 3x 4 = 3x 2 √ 3x 4 + 7 lim x→∞ x2 + 3 Answer √ The limit is 3. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 33 / 37
71. 71. Solution √ 3x 4 + 7 x 4 (3 + 7/x 4 ) lim = lim x→∞ x2 + 3 x→∞ x 2 (1 + 3/x 2 ) x 2 (3 + 7/x 4 ) = lim x→∞ x 2 (1 + 3/x 2 ) (3 + 7/x 4 ) = lim x→∞ 1 + 3/x 2 √ 3+0 √ = = 3. 1+0 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 34 / 37
72. 72. Rationalizing to get a limit Example Compute lim 4x 2 + 17 − 2x . x→∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 35 / 37
73. 73. Rationalizing to get a limit Example Compute lim 4x 2 + 17 − 2x . x→∞ Solution This limit is of the form ∞ − ∞, which we cannot use. So we rationalize the numerator (the denominator is 1) to get an expression that we can use the limit laws on. √ 4x 2 + 17 + 2x lim 4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √ x→∞ x→∞ 4x 2 + 17 + 2x (4x 2 + 17) − 4x 2 = lim √ x→∞ 4x 2 + 17 + 2x 17 = lim √ =0 x→∞ 4x 2 + 17 + 2x V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 35 / 37
74. 74. Kick it up a notch Example Compute lim 4x 2 + 17x − 2x . x→∞ V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 36 / 37
75. 75. Kick it up a notch Example Compute lim 4x 2 + 17x − 2x . x→∞ Solution Same trick, diﬀerent answer: lim 4x 2 + 17x − 2x x→∞ √ 4x 2 + 17x + 2x = lim 4x 2 + 17x − 2x · √ x→∞ 4x 2 + 17x + 2x (4x 2 + 17x) − 4x 2 = lim √ x→∞ 4x 2 + 17x + 2x 17x 17 17 = lim √ = lim = x→∞ 4x 2 + 17x + 2x x→∞ 4 + 17/x + 2 4 V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 36 / 37
76. 76. Summary Inﬁnity is a more complicated concept than a single number. There are rules of thumb, but there are also exceptions. Take a two-pronged approach to limits involving inﬁnity: Look at the expression to guess the limit. Use limit rules and algebra to verify it. V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 37 / 37