1. State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Objective – To be able to find the limit of a function.

2. Definition: And say “the limit of f(x), as x approaches a, equals L” This says that the values of f(x) get closer and closer to the number L as x gets closer to the number a (from either side)

3. Example 1 Find the Limit of f(x) for these three cases: 2 2 2

4. Example 2 Find the Limit of f(x) for these four cases: 2 2 2 DNE

5. Example 3 Find the Limit of f(x) for these four cases: 1 3 DNE 1.5

6. –5–4–3–2–112543 –5 –4 –3 –2 –1 1 2 5 4 3 Example 4 Find the Limit of f(x) for these three cases: - ∞ + ∞ DNE

7. Example 5 Find the Limit of f(x) for these three cases: + ∞

8. Example 6 Find the Limit of f(x) for these two cases: 4

9. Example 7 Find the Limit of f(x) for these two cases: + ∞ -2

10. WS on Limits

11. Pg. 102 4 – 9

12. State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Objective – To be able to find the limit of a function.

13. Example 1 + ∞ 0 DNE

14. Example 2 – ∞ 2 1 2

15. Example 3 0 0 DNE 0 0

16. –5–4–3–2–112543 –5 –4 –3 –2 –1 1 2 5 4 3 DNE 0 + ∞ DNE 0 – ∞

17. State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Objective – To be able to find the limit of a function.

18. Example 1 Use a table of values to estimate the value of the limit. 0.750.900.99?0.9991.0011.011.11.25 x approaches 1 from the LEFT x approaches 1 From the RIGHT = 1

19. -0.1-0.05-0.00100.001 0.050.1 f(x) x Example 2 Use a table of values to estimate the value of the limit. 100 4001x10 6 ? 400100 x approaches +∞ from the LEFT x approaches +∞ From the RIGHT = + ∞

20. 1.951.9951.99922.0012.0152.1 f(x) x Example 3 Use a table of values to estimate the value of the limit. 63.2063.9263.998?64.00164.2465 x approaches 64 from the LEFT x approaches 64 From the RIGHT = 64

21. Pg. 103 15 – 20

22. -1.1-1.05-1.001-0.999 -0.95-0.8 f(x) x Use a table of values to estimate the value of the limit. -9-19-999?1001 216 x approaches –∞ from the LEFT x approaches +∞ From the RIGHT = DNE

23. State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Objective – To be able to find the limit of a function.

24. Example for case 1: Ex for case 2: = 16 = 8 = 2 = -4 Ex for case 3: = 3 2 = 9

25. Limit Laws Suppose that c is a constant: and Sum and Difference: Product: Division: Scalar Mult.:

26. Example 1a Find the Limit: 3(4) – 10 + 4 = 6

27. Example 1b Find the Limit:

28. Example 2 Find the Limit:

29. Pg. 111-112 1 – 9

30. Evaluate the limit. = 3 / 2 = 9

31. State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Objective – To be able to find the limit of a function.

32. Example 1 Find: Functions with the Direct Substitution Property are called continuous at a. However, not all limits can be evaluated by direct substitution, as the following example shows: = 1+1 = 2

33. Example 2 Find the Limit:

34. Example 3 Find the Limit:

35. Example 4 Find the Limit:

36. Example 5 Find the Limit:

37. Pg. 112 11 – 29 odd

38. Evaluate the limit. = DNE = 16

39. State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Objective – To be able to find the limit of a function.

40. If a function f is not continuous at a point c, we say that f is discontinuous at c or c is a point of discontinuity of f.

41. Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0, x=3, and x=4, because the one-sided limits match the value of the function 1234 1 2

42. jump infinite oscillating Essential Discontinuities: Removable Discontinuities: (You can fill the hole.)

43. Removing a discontinuity: has a discontinuity at. Write an extended function that is continuous at. Note: There is another discontinuity at that can not be removed.

44. Removing a discontinuity: Note: There is another discontinuity at that can not be removed.

45. Example Find the value of x which f is not continuous, which of the discontinuities are removable? Removable discontinuity is at: Where as x – 1 is NOT a removable discontinuity.

46. Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

47. WS 1 – 10, 13 – 17 odd Pg. 133 1-6, 10-12, and 15 – 20

48. –1 –5–4–3–2–112543 4 1 2 3 5 6 9 8 7 10 Describe the continuity of the graph.

49. State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Objective – To be able to find the limit of a function.

50. Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either or

51. Case 1: Same Degree Case 2: Degree smaller in Numerator Case 3: Degree smaller in Denominator

52. Example 1 Case 1: Numerator and Denominator of Same Degree Divide numerator and denominator by x 2 00 0

53. Example 2 Case 2: Degree of Numerator Less than Degree of Denominator Divide numerator and denominator by x 3 00 0

54. Example 3 Case 3: Degree of Numerator Greater Than Degree of Denominator Divide numerator and denominator by x 0 0

55. Example 4 a)b) 0 0 0 0 0 0

56. Example 5 Find the Limit:

57. Example 6 Find the Limit:

58. WS 1 – 8 and Pg. 147 11 – 18, and 20 – 22

59. Solve and show work! 00 0

60. State Standard – 4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function. Objective – To be able to find the tangent line.

61. Definition of a Tangent Line: –1 –5–4–3–2–112543 4 1 2 3 5 6 9 8 7 10 P Q Tangent Line

62. Slope: ax P (a,f(a)) Q (x,f(x)) x – a f(x) – f(a)

63. Definition The tangent line to the curve y = f(x) at the point P(a,f(a)) is the line through P with the slope: Provided that this limit exists.

64. Example 1 Find an equation of the tangent line to the parabola y=x 2 at the point (2,4). Use Point Slope y – y 1 = m (x – x 1 ) y – 4 = 4(x – 2) y – 4 = 4x – 8 +4 y = 4x – 4

65. Provided that this limit exists. For many purposes it is desirable to rewrite this expression in an alternative form by letting: h = x – a Then x = a + h

66. Example 2 Find an equation of the tangent line to the hyperbola at the point (3,1). y – y 1 = m (x – x 1 ) y – 1 = - 1 / 3 (x – 3) y – 1 = - 1 / 3 x + 1 +1 y = - 1 / 3 x + 2

67. Example 3 Find an equation of the tangent line to the parabola y = x 2 at the point (3,9). y – y 1 = m (x – x 1 ) y – 9 = 6(x – 3) y – 9 = 6x – 18 +9 y = 6x – 9

68. Example 4 Find an equation of the tangent line to the parabola y = x 2 –4 at the point (1,-3). y – y 1 = m (x – x 1 ) y – -3 = 2(x – 1) y + 3 = 2x – 2 -3 y = 2x – 5

69. Pg. 156 5a, 5b, 6a, 6b, 7 – 10, 11a, 12a, 13b, and 14b

70. State Standard – 4.0 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function. Objective – To be able to find the derivative of a function.

71. is called the derivative of at. We write: “The derivative of f with respect to a is …” There are many ways to write the derivative of

72. “f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks of eff of ecks”or “the derivative of f of x”

73. Example 1 Find the derivative of the function f(x) = x 2 – 8x + 9 at the number ‘a’.

74. Example 2 Find the derivative of the function (x – 9) (x +h – 9)

75. Pg. 163 13 – 17