Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Section 1.3
               The Concept of Limit

                V63.0121.002.2010Su, Calculus I

                        ...
Announcements




           WebAssign Class Key: nyu
           0127 7953
           Office Hours: MR
           5:00–5:4...
Objectives




           Understand and state the
           informal definition of a limit.
           Observe limits on...
Last Time




         Key concept: function
         Properties of functions: domain and range
         Kinds of function...
Limit




        .   .   .   .   .   .
Zeno's Paradox




                                                                   That which is in
                   ...
Outline


 Heuristics


 Errors and tolerances


 Examples


 Pathologies


 Precise Definition of a Limit



            ...
Heuristic Definition of a Limit



 Definition
 We write
                                              lim f(x) = L
      ...
Outline


 Heuristics


 Errors and tolerances


 Examples


 Pathologies


 Precise Definition of a Limit



            ...
The error-tolerance game

 A game between two players to decide if a limit lim f(x) exists.
                              ...
The error-tolerance game




     L
     .




           .
                                            a
                ...
The error-tolerance game




     L
     .




           .
                                            a
                ...
The error-tolerance game




     L
     .




           .
                                            a
                ...
The error-tolerance game


                                                         T
                                    ...
The error-tolerance game




     L
     .




           .
                                            a
                ...
The error-tolerance game


                                                         S
                                    ...
The error-tolerance game




     L
     .




           .
                                            a
                ...
The error-tolerance game


                                                         T
                                    ...
The error-tolerance game


                                                         S
                                    ...
The error-tolerance game




     L
     .




           .
                                            a
                ...
The error-tolerance game




     L
     .




           .
                                            a
                ...
Outline


 Heuristics


 Errors and tolerances


 Examples


 Pathologies


 Precise Definition of a Limit



            ...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0




                            ...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0


 Solution

 Step 1 Player 1: I...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0


 Solution

 Step 1 Player 1: I...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0


 Solution

 Step 1 Player 1: I...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0


 Solution

 Step 1 Player 1: I...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0


 Solution

 Step 1 Player 1: I...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0


 Solution

 Step 1 Player 1: I...
Playing the Error-Tolerance game with x2
 Example
 Find lim x2 if it exists.
         x→0


 Solution

 Step 1 Player 1: I...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Graphical version of the E-T game with x2

                                                    . .
                       ...
Limit of a piecewise function

 Example
          |x|
 Find lim     if it exists.
      x→0 x




                        ...
Limit of a piecewise function

 Example
          |x|
 Find lim     if it exists.
      x→0 x


 Solution
 The function ca...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                              y
                  ...
The E-T game with a piecewise function

                                                                  y
              ...
The E-T game with a piecewise function

                                                                  y
              ...
One-sided limits



 Definition
 We write
                                              lim f(x) = L
                     ...
One-sided limits



 Definition
 We write
                                              lim f(x) = L
                     ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
The error-tolerance game on the right
                                                     y
                             ...
Limit of a piecewise function

 Example
          |x|
 Find lim     if it exists.
      x→0 x


 Solution
 The function ca...
Another Example


 Example
                  1
 Find lim+          if it exists.
         x→0      x




                 ...
The error-tolerance game with lim (1/x)
                                                                x→0
              ...
The error-tolerance game with lim (1/x)
                                                                x→0
              ...
The error-tolerance game with lim (1/x)
                                                                x→0
              ...
The error-tolerance game with lim (1/x)
                                                                x→0
              ...
The error-tolerance game with lim (1/x)
                                                                x→0
              ...
The error-tolerance game with lim (1/x)
                                                                x→0
              ...
The error-tolerance game with lim (1/x)
                                                                x→0
              ...
Another (Bad) Example: Unboundedness


 Example
                  1
 Find lim+          if it exists.
         x→0      x
...
Weird, wild stuff



 Example
                     (π )
 Find lim sin                 if it exists.
         x→0          ...
Function values



           x   π/x  sin(π/x)                                                       . /2
               ...
Weird, wild stuff



 Example
                     (π )
 Find lim sin                 if it exists.
         x→0          ...
Weird, wild stuff



 Example
                     (π )
 Find lim sin                 if it exists.
         x→0          ...
Weird, wild stuff



 Example
                     (π )
 Find lim sin                 if it exists.
         x→0          ...
Weird, wild stuff



 Example
                     (π )
 Find lim sin                 if it exists.
         x→0          ...
Weird, wild stuff



 Example
                     (π )
 Find lim sin                 if it exists.
         x→0          ...
Weird, wild stuff continued

 Here is a graph of the function:
                                                    y
     ...
Outline


 Heuristics


 Errors and tolerances


 Examples


 Pathologies


 Precise Definition of a Limit



            ...
What could go wrong?
Summary of Limit Pathologies




 How could a function fail to have a limit? Some possibilities:
    ...
Meet the Mathematician: Augustin Louis Cauchy




          French, 1789–1857
          Royalist and Catholic
          ma...
Outline


 Heuristics


 Errors and tolerances


 Examples


 Pathologies


 Precise Definition of a Limit



            ...
Precise Definition of a Limit
No, this is not going to be on the test




 Let f be a function defined on an some open int...
The error-tolerance game = ε, δ




      L
      .




           .
                                             a
      ...
The error-tolerance game = ε, δ




  L
  . +ε
    L
    .
  . −ε
  L




           .
                                   ...
The error-tolerance game = ε, δ




  L
  . +ε
    L
    .
  . −ε
  L




           .
                                   ...
The error-tolerance game = ε, δ



                                            T
                                         ...
The error-tolerance game = ε, δ




  L
  . +ε
    L
    .
  . −ε
  L




           .
                                   ...
The error-tolerance game = ε, δ



                                         T
                                         . h...
The error-tolerance game = ε, δ



                                            S
                                         ...
Summary



                                                                                      y
                       ...
Upcoming SlideShare
Loading in …5
×

Lesson 2: The Concept of Limit

1,512 views

Published on

Published in: Education
  • Be the first to comment

Lesson 2: The Concept of Limit

  1. 1. Section 1.3 The Concept of Limit V63.0121.002.2010Su, Calculus I New York University May 18, 2010 Announcements WebAssign Class Key: nyu 0127 7953 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 WebAssign Class Key: nyu 0127 7953 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 1.3 The Concept of Limit May 18, 2010 2 / 32
  3. 3. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 3 / 32
  4. 4. Last Time Key concept: function Properties of functions: domain and range Kinds of functions: linear, polynomial, power, rational, algebraic, transcendental. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 4 / 32
  5. 5. Limit . . . . . .
  6. 6. Zeno's Paradox That which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics VI:9, 239b10) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 5 / 32
  7. 7. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 6 / 32
  8. 8. Heuristic Definition of a 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 1.3 The Concept of Limit May 18, 2010 7 / 32
  9. 9. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 8 / 32
  10. 10. The error-tolerance game A game between two players to decide if a limit lim f(x) exists. x→a Step 1 Player 1 proposes L to be the limit. Step 2 Player 2 chooses an “error” level around L: the maximum amount f(x) can be away from L. Step 3 Player 1 looks for a “tolerance” level around a: the maximum amount x can be from a while ensuring f(x) is within the given error of L. The idea is that points x within the tolerance level of a are taken by f to y-values within the error level of L, with the possible exception of a itself. If Player 1 can do this, he wins the round. If he cannot, he loses the game: the limit cannot be L. Step 4 Player 2 go back to Step 2 with a smaller error. Or, he can give up and concede that the limit is L. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 9 / 32
  11. 11. The error-tolerance game L . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  12. 12. The error-tolerance game L . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  13. 13. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  14. 14. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  15. 15. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  16. 16. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  17. 17. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  18. 18. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  19. 19. 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  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. If Player 2 shrinks the error, Player 1 can still win. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  21. 21. 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. If Player 2 shrinks the error, Player 1 can still win. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  22. 22. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 11 / 32
  23. 23. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  24. 24. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  25. 25. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  26. 26. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  27. 27. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  28. 28. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so a tolerance of 0.01 fits your error of 0.0001. … . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  29. 29. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so a tolerance of 0.01 fits your error of 0.0001. … Can you convince Player 2 that Player 1 can win every round? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  30. 30. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so a tolerance of 0.01 fits your error of 0.0001. … Can you convince Player 2 that Player 1 can win every round? Yes, by setting the tolerance equal to the square root of the error, Player 1 can always win. Player 2 should give up and concede that the limit is 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  31. 31. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  32. 32. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  33. 33. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  34. 34. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  35. 35. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  36. 36. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  37. 37. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  38. 38. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  39. 39. Graphical version of the E-T game with x2 . . y . . . x . . No matter how small an error band Player 2 picks, Player 1 can find a fitting tolerance band. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  40. 40. Limit of a piecewise function Example |x| Find lim if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
  41. 41. Limit of a piecewise function Example |x| Find lim if it exists. x→0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
  42. 42. The E-T game with a piecewise function y . . . . . 1 . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  43. 43. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 1 . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  44. 44. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 1 . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  45. 45. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  46. 46. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x . No. Part of graph inside . 1. − blue is not inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  47. 47. The E-T game with a piecewise function y . . . . . 1 O . h, I guess the limit isn’t 1 . . .. x . No. Part of graph inside . 1. − blue is not inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  48. 48. The E-T game with a piecewise function y . . . . . 1 . think the limit is −1 I . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  49. 49. The E-T game with a piecewise function y . . . . . 1 . think the limit is −1 I . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  50. 50. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  51. 51. The E-T game with a piecewise function y . . . No. Part of . graph inside . . 1 blue is not inside . ow about this for a tolerance? green H . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  52. 52. The E-T game with a piecewise function y . . . No. Part of . graph inside . . 1 blue is not inside . h, I guess the limit isn’t −1 O green . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  53. 53. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 0 . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  54. 54. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 0 . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  55. 55. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  56. 56. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x . No. None of . 1. − graph inside blue is inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  57. 57. The E-T game with a piecewise function y . . . . . 1 . . Oh, I guess the . .. x limit isn’t 0 . No. None of . 1. − graph inside blue is inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  58. 58. The E-T game with a piecewise function y . . . . . 1 . I give up! I . guess there’s . .. x no limit! . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  59. 59. One-sided limits Definition 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
  60. 60. One-sided limits Definition We 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 as we like) by taking x to be sufficiently close to a and less than a. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
  61. 61. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  62. 62. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  63. 63. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  64. 64. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  65. 65. The error-tolerance game on the right y . . . 1 . x . . All of graph in- . 1. − side blue is in- side green . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  66. 66. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  67. 67. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  68. 68. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  69. 69. The error-tolerance game on the right y . . All of graph in- . . 1 side blue is in- side green . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  70. 70. The error-tolerance game on the right y . . All of graph in- . . 1 side blue is in- side green . x . . 1. − So lim+ f(x) = 1 and lim f(x) = −1 x→0 x→0− . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  71. 71. Limit of a piecewise function Example |x| Find lim if it exists. x→0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? The error-tolerance game fails, but lim f(x) = 1 lim f(x) = −1 x→0+ x→0− . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 18 / 32
  72. 72. Another Example Example 1 Find lim+ if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 19 / 32
  73. 73. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  74. 74. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  75. 75. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  76. 76. The error-tolerance game with lim (1/x) x→0 y . . The graph escapes the green, so no good .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  77. 77. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  78. 78. The error-tolerance game with lim (1/x) x→0 y . E . ven worse! .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  79. 79. The error-tolerance game with lim (1/x) x→0 y . . The limit does not ex- ist because the func- tion is unbounded near 0 .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  80. 80. Another (Bad) Example: Unboundedness Example 1 Find lim+ if it exists. x→0 x Solution The limit does not exist because the function is unbounded near 0. Later we will talk about the statement that 1 lim+ = +∞ x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 21 / 32
  81. 81. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 22 / 32
  82. 82. 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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 23 / 32
  83. 83. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  84. 84. Weird, wild stuff Example (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 = . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  85. 85. Weird, wild stuff Example (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 = . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  86. 86. Weird, wild stuff Example (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 = . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  87. 87. Weird, wild stuff Example (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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  88. 88. Weird, wild stuff continued Here 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 25 / 32
  89. 89. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 26 / 32
  90. 90. What could go wrong? Summary of Limit Pathologies How 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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 27 / 32
  91. 91. 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 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 28 / 32
  92. 92. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 29 / 32
  93. 93. Precise Definition of a Limit No, this is not going to be on the test Let f be a function 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. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 30 / 32
  94. 94. The error-tolerance game = ε, δ L . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  95. 95. The error-tolerance game = ε, δ L . +ε L . . −ε L . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  96. 96. The error-tolerance game = ε, δ L . +ε L . . −ε L . . − δ. . + δ a aa . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  97. 97. The error-tolerance game = ε, δ T . his δ is too big L . +ε L . . −ε L . . − δ. . + δ a aa . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  98. 98. The error-tolerance game = ε, δ L . +ε L . . −ε L . . −. δ δ a . a+ a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  99. 99. The error-tolerance game = ε, δ T . his δ looks good L . +ε L . . −ε L . . −. δ δ a . a+ a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  100. 100. The error-tolerance game = ε, δ S . o does this δ L . +ε L . . −ε L . . .− δ δ aa .+ a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  101. 101. Summary y . Fundamental Concept: . . 1 limit Error-Tolerance game . x . gives a methods of arguing limits do or do not exist Limit FAIL: jumps, . 1. − unboundedness, sin(π/x) FAIL . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 32 / 32

×