Copyright © Johns and Bartlett ；滄海書局 CHAPTER 13 Partial Derivatives 13.2 Limits and Continuity.

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Presentation transcript:

Copyright © Johns and Bartlett ；滄海書局 CHAPTER 13 Partial Derivatives 13.2 Limits and Continuity

Copyright © Johns and Bartlett ；滄海書局 2 Terminology

Copyright © Johns and Bartlett ；滄海書局 3

Copyright © Johns and Bartlett ；滄海書局 4 FIGURE Various regions in 2-space

Copyright © Johns and Bartlett ；滄海書局 5 FIGURE Various regions in 2-space

Copyright © Johns and Bartlett ；滄海書局 6 FIGURE Various regions in 2-space

Copyright © Johns and Bartlett ；滄海書局 7 FIGURE Various regions in 2-space

Copyright © Johns and Bartlett ；滄海書局 8 Limits of Functions of Two Variables

Copyright © Johns and Bartlett ；滄海書局 9

Copyright © Johns and Bartlett ；滄海書局 10 FIGURE Three of many ways of approaching the point (a, b)

Copyright © Johns and Bartlett ；滄海書局 11 FIGURE Three of many ways of approaching the point (a, b)

Copyright © Johns and Bartlett ；滄海書局 12

Copyright © Johns and Bartlett ；滄海書局 13

Copyright © Johns and Bartlett ；滄海書局 14

Copyright © Johns and Bartlett ；滄海書局 15 FIGURE Graph of function in Example 2

Copyright © Johns and Bartlett ；滄海書局 16

Copyright © Johns and Bartlett ；滄海書局 17 Properties of Limits

Copyright © Johns and Bartlett ；滄海書局 18

Copyright © Johns and Bartlett ；滄海書局 19

Copyright © Johns and Bartlett ；滄海書局 20 Continuity

Copyright © Johns and Bartlett ；滄海書局 21 FIGURE Function has an infinite discontinuity at (0,0)

Copyright © Johns and Bartlett ；滄海書局 22

Copyright © Johns and Bartlett ；滄海書局 23

Copyright © Johns and Bartlett ；滄海書局 24 FIGURE Graph of function in Example 7

Copyright © Johns and Bartlett ；滄海書局 25 FIGURE Graph of function in Example 7

Copyright © Johns and Bartlett ；滄海書局 26 Polynomial and Rational Funtions FIGURE Three discontinuous functions

Copyright © Johns and Bartlett ；滄海書局 27 FIGURE Three discontinuous functions

Copyright © Johns and Bartlett ；滄海書局 28 FIGURE Three discontinuous functions

Copyright © Johns and Bartlett ；滄海書局 29 Functions of Three or More Variables

Copyright © Johns and Bartlett ；滄海書局 30 Formal Definition of a Limit

Copyright © Johns and Bartlett ；滄海書局 31

Copyright © Johns and Bartlett ；滄海書局 32 FIGURE When (x, y) ≠ (a, b) is in the open disk, f(x, y) is in the interval (L – ε, L + ε)

Copyright © Johns and Bartlett ；滄海書局 33

Copyright © Johns and Bartlett ；滄海書局 34

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