Lesson 3: Limits

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Lesson 3: Limits

  1. 1. Section 1.3 The Concept of Limit V63.0121.041, Calculus I New York University September 13, 2010Announcements Let us know if you bought a WebAssign license last year and cannot login First written HW due Wednesday Get-to-know-you survey and photo deadline is October 1 . . . . . .
  2. 2. Announcements Let us know if you bought a WebAssign license last year and cannot login First written HW due Wednesday Get-to-know-you survey and photo deadline is October 1 . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 2 / 36
  3. 3. Guidelines for written homework Papers should be neat and legible. (Use scratch paper.) Label with name, lecture number (041), recitation number, date, assignment number, book sections. Explain your work and your reasoning in your own words. Use complete English sentences. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 3 / 36
  4. 4. Rubric Points Description of Work 3 Work is completely accurate and essentially 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 ex- plained, 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. Solution is completely in- correct. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 4 / 36
  5. 5. Examples of written homework: Dont . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 5 / 36
  6. 6. Examples of written homework: Do . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 6 / 36
  7. 7. Examples of written homework: DoWritten Explanations . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 7 / 36
  8. 8. Examples of written homework: DoGraphs . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 8 / 36
  9. 9. Objectives Understand and state the informal definition of a limit. Observe limits on a graph. Guess limits by algebraic manipulation. Guess limits by numerical information. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 9 / 36
  10. 10. Limit . . . . . .
  11. 11. Zenos Paradox That which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics VI:9, 239b10) . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 10 / 36
  12. 12. OutlineHeuristicsErrors and tolerancesExamplesPathologiesPrecise Definition of a Limit . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 11 / 36
  13. 13. Heuristic Definition of a LimitDefinitionWe write lim f(x) = L x→aand 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 aswe like) by taking x to be sufficiently close to a (on either side of a) butnot equal to a. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 12 / 36
  14. 14. OutlineHeuristicsErrors and tolerancesExamplesPathologiesPrecise Definition of a Limit . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 13 / 36
  15. 15. The error-tolerance gameA game between two players (Dana and Emerson) to decide if a limitlim f(x) exists.x→aStep 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 counting 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 14 / 36
  16. 16. The error-tolerance game L . . a . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  17. 17. The error-tolerance game L . . a . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  18. 18. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  19. 19. The error-tolerance game T . his tolerance is too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  20. 20. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  21. 21. The error-tolerance game S . till too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  22. 22. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  23. 23. The error-tolerance game T . his looks good L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  24. 24. The error-tolerance game S . o does this L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  25. 25. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. Even if Emerson shrinks the error, Dana can still move. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  26. 26. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. Even if Emerson shrinks the error, Dana can still move. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
  27. 27. OutlineHeuristicsErrors and tolerancesExamplesPathologiesPrecise Definition of a Limit . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 16 / 36
  28. 28. ExampleFind lim x2 if it exists. x→0 . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
  29. 29. ExampleFind lim x2 if it exists. x→0Solution Dana claims the limit is zero. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
  30. 30. ExampleFind lim x2 if it exists. x→0Solution 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
  31. 31. ExampleFind lim x2 if it exists. x→0Solution 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
  32. 32. ExampleFind lim x2 if it exists. x→0Solution 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. If Emerson re-challenges with an error level of 0.0001 = 10−4 , what should Dana’s tolerance be? . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
  33. 33. ExampleFind lim x2 if it exists. x→0Solution 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. 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
  34. 34. ExampleFind lim x2 if it exists. x→0Solution 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. 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 . Dana has a shortcut: By setting tolerance equal to the square root of the error, Dana can win every round. Once Emerson realizes this, Emerson must give up. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
  35. 35. Example |x|Find lim if it exists. x→0 x . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 18 / 36
  36. 36. Example |x|Find lim if it exists. x→0 xSolutionThe function can also be written as { |x| 1 if x > 0; = x −1 if x < 0What would be the limit? . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 18 / 36
  37. 37. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  38. 38. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  39. 39. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  40. 40. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  41. 41. The error-tolerance game y . . . 1 . x . . Part of graph . 1. − inside blue is not inside green . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  42. 42. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  43. 43. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  44. 44. The error-tolerance game y . . Part of graph inside blue is not . . 1 inside green . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  45. 45. The error-tolerance game y . . Part of graph inside blue is not . . 1 inside green . x . . 1. − These are the only good choices; the limit does not exist. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
  46. 46. One-sided limitsDefinitionWe 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 aswe like) by taking x to be sufficiently close to a and greater than a. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 20 / 36
  47. 47. One-sided limitsDefinitionWe write lim f(x) = L x→a−and say “the limit of f(x), as x approaches a from the left, equals L”if we can make the values of f(x) arbitrarily close to L (as close to L aswe like) by taking x to be sufficiently close to a and less than a. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 20 / 36
  48. 48. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  49. 49. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  50. 50. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  51. 51. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  52. 52. The error-tolerance game y . . . 1 . x . . Part of graph . 1. − inside blue is inside green . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  53. 53. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  54. 54. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  55. 55. The error-tolerance game y . . . 1 . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  56. 56. The error-tolerance game y . . Part of graph . . 1 inside blue is inside green . x . . 1. − . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  57. 57. The error-tolerance game y . . Part of graph . . 1 inside blue is inside green . x . . 1. − So lim+ f(x) = 1 and lim f(x) = −1 x→0 x→0− . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
  58. 58. Example |x|Find lim if it exists. x→0 xSolutionThe function can also be written as { |x| 1 if x > 0; = x −1 if x < 0What would be the limit?The error-tolerance game fails, but lim f(x) = 1 lim f(x) = −1 x→0+ x→0− . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 22 / 36
  59. 59. Example 1Find lim+ if it exists. x→0 x . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 23 / 36
  60. 60. The error-tolerance game y . .?. L . x . 0 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
  61. 61. The error-tolerance game y . .?. L . x . 0 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
  62. 62. The error-tolerance game y . .?. L . x . 0 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
  63. 63. The error-tolerance game y . . The graph escapes the green, so no good .?. L . x . 0 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
  64. 64. The error-tolerance game y . .?. L . x . 0 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
  65. 65. The error-tolerance game y . E . ven worse! .?. L . x . 0 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
  66. 66. The error-tolerance game y . . The limit does not ex- ist because the func- tion is unbounded near 0 .?. L . x . 0 . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
  67. 67. Example 1Find lim+ if it exists. x→0 xSolutionThe limit does not exist because the function is unbounded near 0.Next week we will understand the statement that 1 lim+ = +∞ x→0 x . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 25 / 36
  68. 68. Weird, wild stuffExample (π )Find lim sin if it exists. x→0 x . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 26 / 36
  69. 69. Function values x π/x sin(π/x) . /2 π . 1 π 0 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 . . π . .. 0 2/9 9π/2 1 2/13 13π/2 1 2/3 3π/2 −1 2/7 7π/2 −1 . 2/11 11π/2 −1 3 . π/2 . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 27 / 36
  70. 70. Weird, wild stuffExample (π )Find lim sin if it exists. x→0 x . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
  71. 71. Weird, wild stuffExample (π )Find lim sin if it exists. x→0 x f(x) = 0 when x = f(x) = 1 when x = f(x) = −1 when x = . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
  72. 72. Weird, wild stuffExample (π )Find lim sin if it exists. x→0 x 1 f(x) = 0 when x = for any integer k k f(x) = 1 when x = f(x) = −1 when x = . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
  73. 73. Weird, wild stuffExample (π )Find lim sin if it exists. x→0 x 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 = . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
  74. 74. Weird, wild stuffExample (π )Find lim sin if it exists. x→0 x 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
  75. 75. Weird, wild stuff continuedHere is a graph of the function: 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 29 / 36
  76. 76. OutlineHeuristicsErrors and tolerancesExamplesPathologiesPrecise Definition of a Limit . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 30 / 36
  77. 77. What could go wrong?Summary of Limit PathologiesHow could a function fail to have a limit? Some possibilities: left- and right- hand limits exist but are not equal The function is unbounded near a Oscillation with increasingly high frequency near a . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 31 / 36
  78. 78. Meet the Mathematician: Augustin Louis Cauchy French, 1789–1857 Royalist and Catholic made contributions in geometry, calculus, complex analysis, number theory created the definition of limit we use today but didn’t understand it . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 32 / 36
  79. 79. OutlineHeuristicsErrors and tolerancesExamplesPathologiesPrecise Definition of a Limit . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 33 / 36
  80. 80. Precise Definition of a LimitNo, this is not going to be on the testLet f be a function defined on an some open interval that contains thenumber 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→aif for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε. . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 34 / 36
  81. 81. The error-tolerance game = ε, δ L . . a . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
  82. 82. The error-tolerance game = ε, δL. +ε L .. −εL . a . . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
  83. 83. The error-tolerance game = ε, δL. +ε L .. −εL . . − δ. . + δ a aa . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
  84. 84. The error-tolerance game = ε, δ T . his δ is too bigL. +ε L .. −εL . . − δ. . + δ a aa . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
  85. 85. The error-tolerance game = ε, δL. +ε L .. −εL . . −. δ δ a . a+ a . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
  86. 86. The error-tolerance game = ε, δ T . his δ looks goodL. +ε L .. −εL . . −. δ δ a . a+ a . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
  87. 87. The error-tolerance game = ε, δ S . o does this δL. +ε L .. −εL . . .− δ δ aa .+ a . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
  88. 88. Summary: Many perspectives on limits Graphical: L is the value the function “wants to go to” near a Heuristical: f(x) can be made arbitrarily close to L by taking x sufficiently close to a. Informal: the error/tolerance game Precise: if for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε. Algebraic/Formulaic: next time . . . . . . V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 36 / 36

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