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# Lesson 6: Limits Involving Infinity (slides)

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Infinity is a complicated concept, but there are rules for dealing with both limits at infinity and infinite limits.

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### Lesson 6: Limits Involving Infinity (slides)

1. 1. Sec on 1.6 Limits Involving Inﬁnity V63.0121.011: Calculus I Professor Ma hew Leingang New York University February 9, 2011.
2. 2. Announcements Get-to-know-you extra credit due Friday February 11 Quiz 1 is next week in recita on. Covers Sec ons 1.1–1.4
3. 3. Objectives “Intuit” limits involving inﬁnity by eyeballing the expression. Show limits involving inﬁnity by algebraic manipula on and conceptual argument.
4. 4. Recall the deﬁnition of limit Deﬁni on 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.
5. 5. The unboundedness problem y 1 Recall why lim+ doesn’t x→0 x exist. No ma er how thin we draw the strip to the right of x = 0, L? we cannot “capture” the graph inside the box. . x
6. 6. The unboundedness problem y 1 Recall why lim+ doesn’t x→0 x exist. No ma er how thin we draw the strip to the right of x = 0, L? we cannot “capture” the graph inside the box. . x
7. 7. The unboundedness problem y 1 Recall why lim+ doesn’t x→0 x exist. No ma er how thin we draw the strip to the right of x = 0, L? we cannot “capture” the graph inside the box. . x
8. 8. The unboundedness problem y 1 Recall why lim+ doesn’t x→0 x exist. No ma er how thin we draw the strip to the right of x = 0, L? we cannot “capture” the graph inside the box. . x
9. 9. Outline Inﬁnite Limits Ver cal Asymptotes Inﬁnite Limits we Know Limit “Laws” with Inﬁnite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Ra onalizing to get a limit
10. 10. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a.
11. 11. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a. “Large” takes the place of “close to L”.
12. 12. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a. “Large” takes the place of “close to L”.
13. 13. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a. “Large” takes the place of “close to L”.
14. 14. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a. “Large” takes the place of “close to L”.
15. 15. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a. “Large” takes the place of “close to L”.
16. 16. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a. “Large” takes the place of “close to L”.
17. 17. Inﬁnite LimitsDeﬁni on yThe nota on lim f(x) = ∞ x→ameans that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x suﬃciently close . xto a but not equal to a. “Large” takes the place of “close to L”.
18. 18. Negative Inﬁnity Deﬁni on The nota on lim f(x) = −∞ x→a means that the values of f(x) can be made arbitrarily large nega ve (as large as we please) by taking x suﬃciently close to a but not equal to a.
19. 19. Negative Inﬁnity Deﬁni on The nota on lim f(x) = −∞ x→a means that the values of f(x) can be made arbitrarily large nega ve (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 (nega ve) number.
20. 20. Vertical Asymptotes Deﬁni on The line x = a is called a ver cal 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−
21. 21. Inﬁnite Limits we Know y 1 lim+ =∞ x→0 x . x
22. 22. Inﬁnite Limits we Know y 1 lim+ =∞ x→0 x 1 lim− = −∞ x→0 x . x
23. 23. Inﬁnite Limits we Know y 1 lim+ =∞ x→0 x 1 lim− = −∞ x→0 x . x 1 lim 2 = ∞ x→0 x
24. 24. Finding limits at trouble spots Example Let x2 + 2 f(x) = 2 x − 3x + 2 Find lim− f(x) and lim+ f(x) for each a at which f is not con nuous. x→a x→a
25. 25. Finding limits at trouble spots Example Let x2 + 2 f(x) = 2 x − 3x + 2 Find lim− f(x) and lim+ f(x) for each a at which f is not con nuous. x→a x→a Solu on The denominator factors as (x − 1)(x − 2). We can record the signs of the factors on the number line.
26. 26. Use the number line . (x − 1)
27. 27. Use the number line −. 0 + (x − 1) 1
28. 28. Use the number line −. 0 + (x − 1) 1 − 0 + (x − 2) 2
29. 29. Use the number line −. 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x2 + 2)
30. 30. Use the number line −. 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x2 + 2) f(x) 1 2
31. 31. Use the number line −. 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x2 + 2) + f(x) 1 2
32. 32. Use the number line small −. 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x2 + 2) + +∞ f(x) 1 2 lim f(x) = + ∞ x→1−
33. 33. Use the number line small −. 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x2 + 2) + +∞ −∞ f(x) 1 2 lim f(x) = + ∞ x→1− lim f(x) = − ∞ x→1+
34. 34. Use the number line −. 0 + (x − 1) 1 − 0 + (x − 2) 2 + (x2 + 2) + +∞ −∞− f(x) 1 2 lim f(x) = + ∞ x→1− lim f(x) = − ∞ x→1+
35. 35. Use the number line −. 0 + (x − 1) small 1 − 0 + (x − 2) 2 + (x2 + 2) + +∞ −∞−−∞ f(x) 1 2 lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ x→1+
36. 36. Use the number line −. 0 + (x − 1) small 1 − 0 + (x − 2) 2 + (x2 + 2) + +∞ −∞−−∞ +∞ f(x) 1 2 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 + (x2 + 2) + +∞ −∞−−∞ +∞ + f(x) 1 2 lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+
38. 38. In English, now To explain the limit, you can say/write: “As x → 1− , the numerator approaches 3, and the denominator approaches 0 while remaining posi ve. So the limit is +∞.”
39. 39. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . x −1 1 2 3
40. 40. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . x −1 1 2 3
41. 41. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . x −1 1 2 3
42. 42. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . x −1 1 2 3
43. 43. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . x −1 1 2 3
44. 44. Limit Laws (?) with inﬁnite limits Fact The sum of two posi ve or two nega ve inﬁnite limits is inﬁnite. If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then x→a x→a lim (f(x) + g(x)) = −∞. x→a
45. 45. Rules of Thumb with inﬁnite limits Fact ∞+∞=∞ The sum of two posi ve or two nega ve inﬁnite limits is inﬁnite. If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. . x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then x→a x→a lim (f(x) + g(x)) = −∞. x→a
46. 46. Rules of Thumb with inﬁnite limits Fact ∞+∞=∞ −∞ + (−∞) = −∞ The sum of two posi ve or two nega ve inﬁnite limits is inﬁnite. If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. . x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then x→a x→a . lim (f(x) + g(x)) = −∞. x→a
47. 47. Rules of Thumb with inﬁnite limits Fact ∞+∞=∞ −∞ + (−∞) = −∞ The sum of two posi ve or two nega ve inﬁnite limits is inﬁnite. If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. . x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then x→a x→a . lim (f(x) + g(x)) = −∞. x→a Remark We don’t say anything here about limits of the form ∞ − ∞.
48. 48. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The sum of a ﬁnite limit and an inﬁnite limit is inﬁnite. If lim f(x) = L and lim g(x) = ±∞, then x→a x→a lim (f(x) + g(x)) = ±∞. x→a
49. 49. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The sum of a ﬁnite limit and an inﬁnite limit is inﬁnite. . If lim f(x) = L and lim g(x) = ±∞, then x→a x→a lim (f(x) + g(x)) = ±∞. x→a L+∞=∞ L − ∞ = −∞
50. 50. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite limit is not 0. If lim f(x) = L, lim g(x) = ∞, and L > 0, then x→a x→a lim f(x) · g(x) = ∞. x→a
51. 51. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite limit is not 0. If lim f(x) = L, lim g(x) = ∞, and L > 0, then x→a x→a lim f(x) · g(x) = ∞. x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then x→a x→a lim f(x) · g(x) = −∞. x→a
52. 52. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite limit is not 0. { ∞ If lim f(x) = L, lim g(x)· = ∞, and L > if Lthen 0, > 0 L ∞= x→a x→a −∞ if L < 0. lim f(x) · g(x) = ∞. x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then x→a x→a . lim f(x) · g(x) = −∞. x→a
53. 53. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite limit is not 0. If lim f(x) = L, lim g(x) = −∞, and L > 0, then x→a x→a lim f(x) · g(x) = −∞. x→a
54. 54. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite limit is not 0. If lim f(x) = L, lim g(x) = −∞, and L > 0, then x→a x→a lim f(x) · g(x) = −∞. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then x→a x→a lim f(x) · g(x) = ∞. x→a
55. 55. Rules of Thumb with inﬁnite limitsKids, don’t try this at home! Fact The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite limit is not 0. { If lim f(x) = L, lim g(x) = −∞, −∞ L > L > 0 L · (−∞) = and if 0, then x→a x→a ∞ if L < 0. lim f(x) · g(x) = −∞. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then x→a x→a . lim f(x) · g(x) = ∞. x→a
56. 56. Multiplying inﬁnite limitsKids, don’t try this at home! Fact The product of two inﬁnite limits is inﬁnite. If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞. x→a x→a x→a
57. 57. Multiplying inﬁnite limitsKids, don’t try this at home! ∞ ·∞ · ∞ = ∞ (−∞) = −∞ (−∞) · (−∞) = ∞ Fact The product of two inﬁnite limits is inﬁnite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞. x→a x→a x→a
58. 58. Dividing by InﬁnityKids, don’t try this at home! Fact The quo ent of a ﬁnite limit by an inﬁnite limit is zero. f(x) If lim f(x) = L and lim g(x) = ±∞, then lim = 0. x→a x→a x→a g(x)
59. 59. Dividing by InﬁnityKids, don’t try this at home! L =0 Fact ∞ The quo ent of a ﬁnite limit by an inﬁnite limit is zero. f(x) . If lim f(x) = L and lim g(x) = ±∞, then lim = 0. x→a x→a x→a g(x)
60. 60. 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.
61. 61. Indeterminate Limit forms L Limits of the form are indeterminate. There is no rule for 0 evalua ng 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, x→0 x sin(1/x) 0 even in the le - or right-hand sense. There are inﬁnitely many ver cal asymptotes arbitrarily close to 0!
62. 62. More 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.
63. 63. Indeterminate forms are like Tug Of War Which side wins depends on which side is stronger.
64. 64. Outline Inﬁnite Limits Ver cal Asymptotes Inﬁnite Limits we Know Limit “Laws” with Inﬁnite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Ra onalizing to get a limit
65. 65. Limits at Inﬁnity Deﬁni on Let f be a func on 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.
66. 66. Horizontal Asymptotes Deﬁni on 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→−∞
67. 67. Horizontal Asymptotes Deﬁni on 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!
68. 68. Basic limits at inﬁnity Theorem Let n be a posi ve integer. Then 1 lim n = 0 x→∞ x 1 lim n = 0 x→−∞ x
69. 69. Limit laws at inﬁnity Fact Any limit law that concerns ﬁnite limits at a ﬁnite point a is s ll true if the ﬁnite point is replaced by ±∞. That is, if lim f(x) = L and x→∞ lim g(x) = M, then x→∞ lim (f(x) + g(x)) = L + M x→∞
70. 70. Limit laws at inﬁnity Fact Any limit law that concerns ﬁnite limits at a ﬁnite point a is s ll true if the ﬁnite point is replaced by ±∞. That is, if lim f(x) = L and x→∞ lim g(x) = M, then x→∞ lim (f(x) + g(x)) = L + M x→∞ lim (f(x) − g(x)) = L − M x→∞
71. 71. Limit laws at inﬁnity Fact Any limit law that concerns ﬁnite limits at a ﬁnite point a is s ll true if the ﬁnite point is replaced by ±∞. That is, if lim f(x) = L and x→∞ lim g(x) = M, then x→∞ lim (f(x) + g(x)) = L + M x→∞ lim (f(x) − g(x)) = L − M x→∞ lim cf(x) = c · L (for any x→∞ constant c)
72. 72. Limit laws at inﬁnity Fact Any limit law that concerns ﬁnite limits at a ﬁnite point a is s ll true if the ﬁnite point is replaced by ±∞. That is, if lim f(x) = L and x→∞ lim g(x) = M, then x→∞ lim (f(x) + g(x)) = L + M lim f(x) · g(x) = L · M x→∞ x→∞ lim (f(x) − g(x)) = L − M x→∞ lim cf(x) = c · L (for any x→∞ constant c)
73. 73. Limit laws at inﬁnity Fact Any limit law that concerns ﬁnite limits at a ﬁnite point a is s ll true if the ﬁnite point is replaced by ±∞. That is, if lim f(x) = L and x→∞ lim g(x) = M, then x→∞ lim (f(x) + g(x)) = L + M lim f(x) · g(x) = L · M x→∞ x→∞ lim (f(x) − g(x)) = L − M f(x) L x→∞ lim = (if M ̸= 0) lim cf(x) = c · L (for any x→∞ g(x) M x→∞ constant c)
74. 74. Limit laws at inﬁnity Fact Any limit law that concerns ﬁnite limits at a ﬁnite point a is s ll true if the ﬁnite point is replaced by ±∞. That is, if lim f(x) = L and x→∞ lim g(x) = M, then x→∞ lim (f(x) + g(x)) = L + M lim f(x) · g(x) = L · M x→∞ x→∞ lim (f(x) − g(x)) = L − M f(x) L x→∞ lim = (if M ̸= 0) lim cf(x) = c · L (for any x→∞ g(x) M x→∞ constant c) etc.
75. 75. Computing limits at ∞With the limit laws Example x Find lim if it exists. x→∞ x2 + 1
76. 76. Computing limits at ∞With the limit laws Example x Find lim if it exists. x→∞ x2 + 1 Answer The limit is 0. No ce that the graph does cross the y asymptote, which contradicts one of the . commonly held beliefs of x what an asymptote is.
77. 77. Solution Solu on Factor out the largest power of x from the numerator and denominator. We have x x(1) 1 1 = 2 = · , so x2 + 1 x (1 + 1/x2 ) x 1 + 1/x2 x 1 1 1 1 1 lim 2 = lim = lim · lim =0· = 0. x→∞ x + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2 1+0
78. 78. Another Example Example x3 + 2x2 + 4 Find lim if it exists. x→∞ 3x2 + 1
79. 79. Another Example Example x3 + 2x2 + 4 Find lim if it exists. x→∞ 3x2 + 1 Answer The limit is ∞.
80. 80. Solution Solu on Factor out the largest power of x from the numerator and denominator. We have x3 + 2x2 + 4 x3 (1 + 2/x + 4/x3 ) 1 + 2/x + 4/x3 = =x· , so 3x2 + 1 x2 (3 + 1/x2 ) 3 + 1/x2 ( ) ( ) 1 x3 + 2x2 + 4 1 + 2/x + 4/x3 lim = lim x · = lim x · = ∞ x→∞ 3x2 + 1 x→∞ 3 + 1/x2 x→∞ 3
81. 81. Solution Solu on Factor out the largest power of x from the numerator and denominator. We have x3 + 2x2 + 4 x3 (1 + 2/x + 4/x3 ) 1 + 2/x + 4/x3 = =x· , so 3x2 + 1 x2 (3 + 1/x2 ) 3 + 1/x2 ( ) ( ) 1 x3 + 2x2 + 4 1 + 2/x + 4/x3 lim = lim x · = lim x · = ∞ x→∞ 3x2 + 1 x→∞ 3 + 1/x2 x→∞ 3
82. 82. Yet Another Example Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞
83. 83. Yet Another Example Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞
84. 84. Solution Solu on Factor out the largest power of x from the numerator and denominator. We have 2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 ) = 3 , so 4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 ) 2x3 + 3x + 1 2 + 3/x2 + 1/x3 2+0+0 1 lim 3 2+7 = lim = = x→∞ 4x + 5x x→∞ 4 + 5/x + 7/x3 4+0+0 2
85. 85. Solution Solu on Factor out the largest power of x from the numerator and denominator. We have 2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 ) = 3 , so 4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 ) 2x3 + 3x + 1 2 + 3/x2 + 1/x3 2+0+0 1 lim 3 2+7 = lim = = x→∞ 4x + 5x x→∞ 4 + 5/x + 7/x3 4+0+0 2
86. 86. Upshot of the last three examples Upshot When ﬁnding limits of algebraic expressions at inﬁnity, look at the highest degree terms. If the higher degree is in the numerator, the limit is ±∞. If the higher degree is in the denominator, the limit is 0. If the degrees are the same, the limit is the ra o of the top-degree coeﬃcients.
87. 87. Still Another Example Example Find √ 3x4 + 7 lim x→∞ x2 + 3
88. 88. Still Another Example √ √ √ 3x4 + 7 ∼ 3x4 = 3x2 Example Find √ 3x4 + 7 . lim x→∞ x2 + 3 Answer √ The limit is 3.
89. 89. Solution Solu on √ √ √ 3x4 + 7 x4 (3 + 7/x4 ) x2 (3 + 7/x4 ) lim = lim 2 = lim 2 x→∞ x2 + 3 x→∞ x (1 + 3/x2 ) x→∞ x (1 + 3/x2 ) √ √ (3 + 7/x4 ) 3+0 √ = lim = = 3. x→∞ 1 + 3/x2 1+0
90. 90. Rationalizing to get a limit Example (√ ) Compute lim 4x 2 + 17 − 2x . x→∞
91. 91. Rationalizing to get a limit Example (√ ) Compute lim 4x 2 + 17 − 2x . x→∞ Solu on This limit is of the form ∞ − ∞, which we cannot use. So we ra onalize the numerator (the denominator is 1) to get an expression that we can use the limit laws on. (√ ) (√ ) √4x2 + 17 + 2x lim 4x2 + 17 − 2x = lim + 17 − 2x · √ 4x2 x→∞ x→∞ 4x2 + 17 + 2x (4x + 17) − 4x 2 2 17 = lim √ = lim √ =0 x→∞ 4x2 + 17 + 2x x→∞ 4x2 + 17 + 2x
92. 92. Kick it up a notch Example (√ ) Compute lim 4x 2 + 17x − 2x . x→∞
93. 93. Kick it up a notch Example (√ ) Compute lim 4x 2 + 17x − 2x . x→∞ Solu on Same trick, diﬀerent answer: (√ ) lim 4x2 + 17x − 2x x→∞ (√ ) √4x2 + 17x + 2x = lim 4x2 + 17x − 2x · √ x→∞ 4x2 + 17x + 2x (4x2 + 17x) − 4x2 17x 17 17 = lim √ = lim √ = lim √ = x→∞ 4x2 + 17x + 2x x→∞ 4x2 + 17x + 2x x→∞ 4 + 17/x + 2 4
94. 94. Summary Inﬁnity is a more complicated concept than a single number. There are rules of thumb, but there are also excep ons. 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.