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Limit and Continuity

Limit and Continuity. y. 2.1 Rate of Change and Limits (1) Average and Instantaneous Speed. y. t=2. v=?. 2.1 Rate of Change and Limits (2, Example 2) Average and Instantaneous Speed. When different value of h. y. e. L. e. d. d. x. c.

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Limit and Continuity

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  1. Limit and Continuity

  2. y 2.1 Rate of Change and Limits (1)Average and Instantaneous Speed

  3. y t=2 v=? 2.1 Rate of Change and Limits (2, Example 2)Average and Instantaneous Speed When different value of h

  4. y e L e d d x c 2.1 Rate of Change and Limits (3)Definition of Limit

  5. 2.1 Rate of Change and Limits (4)Definition of Limit

  6. 2.1 Rate of Change and Limits (5)Properties of Limit This can be applied to do the limits of all polynominal and rational functions.

  7. 2.1 Rate of Change and Limits (6, Theorem 1)Properties of Limit

  8. 2.1 Rate of Change and Limits (7, Theorem 1)Properties of Limit

  9. 2.1 Rate of Change and Limits (8, Theorem 1)Properties of Limit

  10. 2.1 Rate of Change and Limits (9, Example 3)Properties of Limit

  11. 2.1 Rate of Change and Limits (10,Theorem 2)Properties of Limit

  12. 2.1 Rate of Change and Limits (11, Example 4)Properties of Limit

  13. 2-1 Exercise 63 2.1 Rate of Change and Limits (12, Example 5)Properties of Limit

  14. 2.1 Rate of Change and Limits (13, Example 6)Properties of Limit

  15. f(x) f(c+) f(c-) x c 2.1 Rate of Change and Limits (14)One-sided and Two-sided Limits

  16. 2.1 Rate of Change and Limits (15, Example 7)One-sided and Two-sided Limits

  17. 2.1 Rate of Change and Limits (16, Theorem 3)One-sided and Two-sided Limits

  18. 2.1 Rate of Change and Limits (17, Example 8)One-sided and Two-sided Limits

  19. y h L f g x c 2.1 Rate of Change and Limits (18, Theorem 4)Sandwich Theorem

  20. h(x) = x2 g(x) = -x2 f(x) = x2 sin(1/x) 2.1 Rate of Change and Limits (19, Example 9) Sandwich Theorem

  21. 2.2 Limits Involving Infinite (1) Finite Limits as x→± • The symbol of infinite () does not represent a real number. • The use  to describe the behavior of a function when the values in its domain or range out grow all finite bounds.

  22. 2.2 Limits Involving Infinite (2) Finite Limits as x→±

  23. 2.2 Limits Involving Infinite (3) Finite Limits as x→±

  24. 2.2 Limits Involving Infinite (4, Example 1)Finite Limits as x→±

  25. 2.2 Limits Involving Infinite (5, Example 2)Sandwich Theorem Revisited

  26. 2.2 Limits Involving Infinite (6, Theorem 5-1)Sandwich Theorem Revisited

  27. 2.2 Limits Involving Infinite (7, Theorem 5-2)Sandwich Theorem Revisited

  28. 2.2 Limits Involving Infinite (8, Example 3)Sandwich Theorem Revisited

  29. 2.2 Limits Involving Infinite (9, Exploration 1-1) Sandwich Theorem Revisited

  30. 2.2 Limits Involving Infinite (10, Exploration 1-2) Sandwich Theorem Revisited

  31. 2.2 Limits Involving Infinite (11, Exploration 1-3) Sandwich Theorem Revisited

  32. 2.2 Limits Involving Infinite (12) Infinite Limits as x →a

  33. 2.2 Limits Involving Infinite (13) Infinite Limits as x →a

  34. 2.2 Limits Involving Infinite (14, Example 4) Infinite Limits as x →a

  35. 2.2 Limits Involving Infinite (15, Example 5) Infinite Limits as x →a

  36. 2.2 Limits Involving Infinite (16, Example 6) End Behavior Models

  37. 2.2 Limits Involving Infinite (17) End Behavior Models

  38. 2.2 Limits Involving Infinite (18, Example 7) End Behavior Models

  39. 2.2 Limits Involving Infinite (19) End Behavior Models • IF one function provides both a left and right behavior model, it called an end behavior model. • In general, g(x) = anxn ia an end behavior model for the polynominal function f(x) = anxn + an-1xn-1 +…+ ao .In the large, all polynominals behave like monomials. • This is the key to the end behavior of rational functions.

  40. 2.2 Limits Involving Infinite (20, Example 8)End Behavior Models

  41. 2.2 Limits Involving Infinite (21, Example 9)End Behavior Models

  42. 2.2 Limits Involving Infinite (22, Example 10)Seeing Limits as x→±

  43. 2.3 Continuity (1, Example 1-1)Continuity at a Point

  44. 2.3 Continuity (2, Example 1-2)Continuity at a Point

  45. Continuity from both side Continuity from the left Continuity from the right a b c 2.3 Continuity (3)Continuity at a Point

  46. 2.3 Continuity (4, Example 2)Continuity at a Point

  47. y = f(x) 2 y = f(x) 1 1 continuous at x=0 continuous at x=0 If it had f(0)=1 y = f(x) 1 continuous at x=0 If it had f(0)=1 2.3 Continuity (5)Continuity at a Point continuity at x = 0 are removable

  48. 1 y = f(x) 2.3 Continuity (6)Continuity at a Point

  49. 2.3 Continuity (7)Continuity at a Point

  50. 2.3 Continuity (8, Exploration1-1,2)Continuity at a Point

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