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# Lesson 3: The Limit of a Function (slides)

The limit is the mathematical formulation of infinitesimal closeness.

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### Lesson 3: The Limit of a Function (slides)

1. 1. Sec on 1.3 The Limit of a Func on V63.0121.011: Calculus I Professor Ma hew Leingang New York University January 31, 2011 Announcements First wri en HW due Wednesday February 2. Get-to-know-you survey and photo deadline is February 11
2. 2. Announcements First wri en HW due Wednesday February 2 Get-to-know-you survey and photo deadline is February 11
3. 3. Guidelines for written homework Papers should be neat and legible. (Use scratch paper.) Label with name, lecture number (011), recita on number, date, assignment number, book sec ons. Explain your work and your reasoning in your own words. Use complete English sentences.
4. 4. Rubric Points Descrip on of Work 3 Work is completely accurate and essen ally perfect. Work is thoroughly developed, neat, and easy to read. Complete sentences are used. 2 Work is good, but incompletely developed, hard to read, unexplained, or jumbled. Answers which are not explained, even if correct, will generally receive 2 points. Work contains “right idea” but is ﬂawed. 1 Work is sketchy. There is some correct work, but most of work is incorrect. 0 Work minimal or non-existent. Solu on is completely incorrect.
5. 5. Written homework: Don’t
6. 6. Written homework: Do
7. 7. Written homework: DoWritten Explanations
8. 8. Written homework: DoGraphs
9. 9. Objectives Understand and state the informal deﬁni on of a limit. Observe limits on a graph. Guess limits by algebraic manipula on. Guess limits by numerical informa on.
10. 10. Limit.
11. 11. Yoda on teaching course concepts You must unlearn what you have learned. In other words, we are building up concepts and allowing ourselves only to speak in terms of what we personally have produced.
12. 12. Zeno’s Paradox That which is in locomo on must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics VI:9, 239b10)
13. 13. Outline Heuris cs Errors and tolerances Examples Precise Deﬁni on of a Limit
14. 14. Heuristic Deﬁnition of a 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.
15. 15. Outline Heuris cs Errors and tolerances Examples Precise Deﬁni on of a Limit
16. 16. The error-tolerance game A game between two players (Dana and Emerson) to decide if a limit lim f(x) exists. x→a Step 1 Dana proposes L to be the limit.
17. 17. The error-tolerance game A game between two players (Dana and Emerson) to decide if a limit lim f(x) exists. x→a Step 1 Dana proposes L to be the limit. Step 2 Emerson challenges with an “error” level around L.
18. 18. The error-tolerance game A game between two players (Dana and Emerson) to decide if a limit lim f(x) exists. x→a Step 1 Dana proposes L to be the limit. Step 2 Emerson challenges with an “error” level around L. Step 3 Dana chooses a “tolerance” level around a so that points x within that tolerance of a (not coun ng a itself) are taken to values y within the error level of L. If Dana cannot, Emerson wins and the limit cannot be L.
19. 19. The error-tolerance game A game between two players (Dana and Emerson) to decide if a limit lim f(x) exists. x→a Step 1 Dana proposes L to be the limit. Step 2 Emerson challenges with an “error” level around L. Step 3 Dana chooses a “tolerance” level around a so that points x within that tolerance of a (not coun ng a itself) are taken to values y within the error level of L. If Dana cannot, Emerson wins and the limit cannot be L. Step 4 If Dana’s move is a good one, Emerson can challenge again or give up. If Emerson gives up, Dana wins and the limit is L.
20. 20. The error-tolerance game L . a
21. 21. The error-tolerance game L . a
22. 22. The error-tolerance game L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip.
23. 23. The error-tolerance game This tolerance is too big L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip.
24. 24. The error-tolerance game L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip.
25. 25. The error-tolerance game S ll too big L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip.
26. 26. The error-tolerance game L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip.
27. 27. The error-tolerance game This looks good L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip.
28. 28. The error-tolerance game So does this L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip.
29. 29. The error-tolerance game L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip. Even if Emerson shrinks the error, Dana can s ll move.
30. 30. The error-tolerance game L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip. Even if Emerson shrinks the error, Dana can s ll move.
31. 31. Outline Heuris cs Errors and tolerances Examples Precise Deﬁni on of a Limit
32. 32. Playing the E-T Game Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on
33. 33. Playing the E-T Game Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on Dana claims the limit is zero.
34. 34. Playing the E-T Game Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on Dana claims the limit is zero. If Emerson challenges with an error level of 0.01, Dana needs to guarantee that −0.01 < x2 < 0.01 for all x suﬃciently close to zero.
35. 35. Playing the E-T Game Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on Dana claims the limit is zero. If Emerson challenges with an error level of 0.01, Dana needs to guarantee that −0.01 < x2 < 0.01 for all x suﬃciently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
36. 36. Playing the E-T Game Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on If Emerson re-challenges with an error level of 0.0001 = 10−4 , what should Dana’s tolerance be?
37. 37. Playing the E-T Game Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on If Emerson re-challenges with an error level of 0.0001 = 10−4 , what should Dana’s tolerance be? A tolerance of 0.01 works because |x| < 10−2 =⇒ x2 < 10−4 .
38. 38. Playing the E-T Game Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on Dana has a shortcut: By se ng tolerance equal to the square root of the error, Dana can win every round. Once Emerson realizes this, Emerson must give up.
39. 39. Graphical version of E-T gamewith x2 y . x
40. 40. Graphical version of E-T gamewith x2 y . x
41. 41. Graphical version of E-T gamewith x2 y . x
42. 42. Graphical version of E-T gamewith x2 y . x
43. 43. Graphical version of E-T gamewith x2 y . x
44. 44. Graphical version of E-T gamewith x2 y . x
45. 45. Graphical version of E-T gamewith x2 y . x
46. 46. Graphical version of E-T gamewith x2 y . x
47. 47. Graphical version of E-T gamewith x2 y No ma er how small an error Emerson picks, Dana can ﬁnd a ﬁ ng tolerance band. . x
48. 48. A piecewise-deﬁned function Example |x| Find lim if it exists. x→0 x
49. 49. A piecewise-deﬁned function Example |x| Find lim if it exists. x→0 x Solu on The func on can also be wri en as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit?
50. 50. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 . x −1
51. 51. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 I think the limit is 1 . x −1
52. 52. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 I think the limit is 1 . x Can you ﬁt an error of 0.5? −1
53. 53. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 How about this for a tol- . x erance? −1
54. 54. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 How about this for a tol- . No. Part of x erance? graph inside −1 blue is not inside green
55. 55. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y Oh, I guess 1 the limit isn’t 1 . No. Part of x graph inside −1 blue is not inside green
56. 56. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 I think the limit is −1 . x −1
57. 57. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 I think the limit is −1 . Can you ﬁt xan error of 0.5? −1
58. 58. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 How about . Can you ﬁt xan this for a tol- error of 0.5? erance? −1
59. 59. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y No. Part of graph inside 1 blue is not How about inside green this for a tol- . x erance? −1
60. 60. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y No. Part of graph inside Oh, I guess 1 blue is not the limit isn’t inside green −1 . x −1
61. 61. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 I think the limit is 0 . x −1
62. 62. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 I think the limit is 0 . Can you ﬁt xan error of 0.5? −1
63. 63. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 How about . Can you ﬁt xan this for a tol- error of 0.5? erance? −1
64. 64. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 How about this for a tol- . No. None of x erance? graph inside −1 blue is inside green
65. 65. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 Oh, I guess the limit isn’t . No. None of x 0 graph inside −1 blue is inside green
66. 66. The E-T game with a piecewisefunction |x| Find lim if it exists. x→0 x y 1 I give up! I guess there’s . x no limit! −1
67. 67. One-sided limits Deﬁni on We write lim f(x) = L x→a+ and say “the limit of f(x), as x approaches a from the right, 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 and greater than a.
68. 68. One-sided limits Deﬁni on We write lim f(x) = L x→a− and say “the limit of f(x), as x approaches a from the le , 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 and less than a.
69. 69. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x −1
70. 70. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x −1
71. 71. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x −1
72. 72. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x −1
73. 73. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x Part of graph −1 inside blue is inside green
74. 74. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x −1
75. 75. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x −1
76. 76. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y 1 . x −1
77. 77. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y Part of graph 1 inside blue is inside green . x −1
78. 78. The error-tolerance game |x| |x| Find lim+ and lim− if they exist. x→0 x x→0 x y Part of graph 1 inside blue is inside green . x −1
79. 79. A piecewise-deﬁned function Example |x| Find lim if it exists. x→0 x Solu on The error-tolerance game fails, but lim f(x) = 1 lim f(x) = −1 x→0+ x→0−
80. 80. Another Example Example 1 Find lim+ if it exists. x→0 x
81. 81. The error-tolerance game with 1/x y 1Find lim+ if it exists. L? x→0 x . x 0
82. 82. The error-tolerance game with 1/x y 1Find lim+ if it exists. L? x→0 x . x 0
83. 83. The error-tolerance game with 1/x y 1Find lim+ if it exists. L? x→0 x . x 0
84. 84. The error-tolerance game with 1/x y The graph escapes the green, so no good 1Find lim+ if it exists. L? x→0 x . x 0
85. 85. The error-tolerance game with 1/x y 1Find lim+ if it exists. L? x→0 x . x 0
86. 86. The error-tolerance game with 1/x y Even worse! 1Find lim+ if it exists. L? x→0 x . x 0
87. 87. The error-tolerance game with 1/x y The limit does not exist because the func on is unbounded near 0 1Find lim+ if it exists. L? x→0 x . x 0
88. 88. Another Example Example 1 Find lim+ if it exists. x→0 x Solu on The limit does not exist because the func on is unbounded near 0. Next week we will understand the statement that 1 lim+ = +∞ x→0 x
89. 89. Weird, wild stuﬀ Example (π ) Find lim sin if it exists. x→0 x
90. 90. Function values x π/x sin(π/x) π/2 π . 0 3π/2
91. 91. Function values x π/x sin(π/x) π 0 π/2 π . 0 3π/2
92. 92. Function values x π/x sin(π/x) π 0 π/2 2π 0 π . 0 3π/2
93. 93. Function values x π/x sin(π/x) π 0 π/2 2π 0 kπ 0 π . 0 3π/2
94. 94. Function values x π/x sin(π/x) 1 π 0 π/2 2π 0 kπ 0 π . 0 3π/2
95. 95. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 kπ 0 π . 0 3π/2
96. 96. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 π . 0 3π/2
97. 97. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 π/2 1 π . 0 3π/2
98. 98. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 π/2 1 5π/2 1 π . 0 3π/2
99. 99. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 π/2 1 5π/2 1 π . 0 9π/2 1 3π/2
100. 100. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 π/2 1 5π/2 1 π . 0 9π/2 1 ((4k + 1)π)/2 1 3π/2
101. 101. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 5π/2 1 π . 0 9π/2 1 ((4k + 1)π)/2 1 3π/2
102. 102. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 9π/2 1 ((4k + 1)π)/2 1 3π/2
103. 103. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 ((4k + 1)π)/2 1 3π/2
104. 104. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 2/(4k + 1) ((4k + 1)π)/2 1 3π/2
105. 105. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 2/(4k + 1) ((4k + 1)π)/2 1 3π/2 −1 3π/2
106. 106. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 2/(4k + 1) ((4k + 1)π)/2 1 3π/2 −1 7π/2 −1 3π/2
107. 107. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 2/(4k + 1) ((4k + 1)π)/2 1 3π/2 −1 7π/2 −1 3π/2 ((4k − 1)π)/2 −1
108. 108. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 2/(4k + 1) ((4k + 1)π)/2 1 2/3 3π/2 −1 7π/2 −1 3π/2 ((4k − 1)π)/2 −1
109. 109. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 2/(4k + 1) ((4k + 1)π)/2 1 2/3 3π/2 −1 2/7 7π/2 −1 3π/2 ((4k − 1)π)/2 −1
110. 110. Function values x π/x sin(π/x) 1 π 0 π/2 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 π . 0 2/9 9π/2 1 2/(4k + 1) ((4k + 1)π)/2 1 2/3 3π/2 −1 2/7 7π/2 −1 3π/2 2/(4k − 1) ((4k − 1)π)/2 −1
111. 111. Weird, wild stuﬀ Example (π ) Find lim sin if it exists. x→0 x
112. 112. Weird, wild stuﬀ Example (π ) Find lim sin if it exists. x→0 x Solu on f(x) = 0 when x = f(x) = 1 when x = f(x) = −1 when x =
113. 113. Weird, wild stuﬀ Example (π ) Find lim sin if it exists. x→0 x Solu on 1 f(x) = 0 when x = for any integer k k f(x) = 1 when x = f(x) = −1 when x =
114. 114. Weird, wild stuﬀ Example (π ) Find lim sin if it exists. x→0 x Solu on 1 f(x) = 0 when x = for any integer k k 2 f(x) = 1 when x = for any integer k 4k + 1 f(x) = −1 when x =
115. 115. Weird, wild stuﬀ Example (π ) Find lim sin if it exists. x→0 x Solu on 1 f(x) = 0 when x = for any integer k k 2 f(x) = 1 when x = for any integer k 4k + 1 2 f(x) = −1 when x = for any integer k 4k − 1
116. 116. Graph Here is a graph of the func on: y 1 . x −1 There are inﬁnitely many points arbitrarily close to zero where f(x) is 0, or 1, or −1. So the limit cannot exist.
117. 117. What could go wrong?Summary of Limit Pathologies How could a func on fail to have a limit? Some possibili es: le - and right- hand limits exist but are not equal The func on is unbounded near a Oscilla on with increasingly high frequency near a
118. 118. Meet the MathematicianAugustin Louis Cauchy French, 1789–1857 Royalist and Catholic made contribu ons in geometry, calculus, complex analysis, number theory created the deﬁni on of limit we use today but didn’t understand it
119. 119. Outline Heuris cs Errors and tolerances Examples Precise Deﬁni on of a Limit
120. 120. Precise Deﬁnition of a LimitNo, this is not going to be on the test Let f be a func on deﬁned on an some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write lim f(x) = L, x→a if for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε.
121. 121. The error-tolerance game = ε, δ L . a
122. 122. The error-tolerance game = ε, δ L+ε L L−ε . a
123. 123. The error-tolerance game = ε, δ L+ε L L−ε . a−δ a a+δ
124. 124. The error-tolerance game = ε, δ This δ is too big L+ε L L−ε . a−δ a a+δ
125. 125. The error-tolerance game = ε, δ L+ε L L−ε . a−δ a a+δ
126. 126. The error-tolerance game = ε, δ This δ looks good L+ε L L−ε . a−δ a a+δ
127. 127. The error-tolerance game = ε, δ So does this δ L+ε L L−ε . a−δ aa+δ
128. 128. SummaryMany perspectives on limits Graphical: L is the value the func on “wants to go to” near a y Heuris cal: f(x) can be made arbitrarily 1 close to L by taking x suﬃciently close to a. . x Informal: the error/tolerance game Precise: if for every ε > 0 there is a −1 corresponding δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε. Algebraic: next me FAIL