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.

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