Week 2 Calculus I PPT - Limits and Continuity

Calculus 1 Lecture on Limits and Continuity (Based on OpenStax Volume 1)

1. Calculus-1 Chapter 1: Limits and Continuity

2. Week 2 In this module we cover 1.1 A Preview of Calculus 1.2 The Limit of a Function  1.3 The Limit Laws 

3. Class Activity: In this section, you will need to complete Desmos activity Limits and Continuity in; https://student.desmos.com Key: X9NYPY 1.1 A Preview of Calculus

4. Secant and Tangent lines The Secant to the function through the points and is the line passing through these points. Its slope is given by The accuracy of approximating the rate of change of the function with a secant line depends on how close is to . As we see in figure, if is closer to , the slope of the secant line is a better measure of the rate of change of at .  

5. Conti. Explanation: As x gets closer to a, the slope of the secant line becomes a better approximation to the rate of change of the function f(x) at a.

6. Secant Lines vs. Tangent Lines A secant line is a straight line joining two points on a function. The black line connecting points P & Q is a secant line. The line is a tangent line to at point P, because it touches it at that single point and follows the direction of the function at the point  

7. Secant and tangent lines GeoGebra Exercise Relationships between Secant and Tangent lines https://www.geogebra.org/m/gsvqhnxf https://www.geogebra.org/m/YvaFWUzY

8. Average Velocity Definition: For an object moving on a straight line with position function , the average velocity of the object on the interval from to , denoted , is given by the formula =  

9. Average and Instantaneous Velocity:

10. Lab Activity Consider the previous activity where the height of a ball at time (in seconds) is given by . Find the average velocity over the given intervals [0.5, 1], [0.5,0.75], [0.5, 0.6], [0.5, 0.55], [0.5,0.55]  

11. 1.2 The Limit of a function Informal Definition of Limit

12. Example: What does the value of approach as approaches ? From left? From right?  

13. Formal Definition of Limits Let be a function defined at all values in an open interval co ntaining a, with the possible exception of a itself and let L be a real number. If all values of the function approach the real number L as the value of approach the number then we say that the limit of as approaches is L. (More succinct, as gets closer to , gets closer and says close to L.) Symbolically,  

14. General Form of finding limit numerically

15. Example Find Explanation: and from the table above we have the following  

16. Class Activity: Construct a table of values to investigate the following limit  

17. Existence of a limit Definition: Let be a function defined at all values in an open interval containing a, with the possible exception of a itself, and Let be a real number. If all values of the function f(x) approach the real number as the values of ) approach the number , then we say that the limit of as approaches a is (More succinct, as x gets closer to gets closer and stays close to ) Symbolically, we express this idea as  

18. One-sided limits Definition limit from the left : Let be a function defined at all values in an open interval of the form , and let be a real number. If the values of the function approach the real number as the values of (where ) approach the number , then we say that is the limit of as approaches from the left. Symbolically, we express this idea as  

19. One-sided limits Definition limits from the right : Let be a function defined at all values in an open interval of the form , and let be a real number. If the values of the function approach the real number as the values of (where ) approach the number , then we say that is the limit of as approaches from the right. Symbolically, we express this idea as  

20. Find (a) (b)   Example:

21. Explanation: This means limit of f as x approaches 2 from the left. i.e consider values of f for values of x<2 (for example, 1.9, 1.99, 1.999) This means limit of f as x approaches 2 from the right. i.e consider values of f for values of x>2 (for example, 2.1, 2.01, 2.001,2.0001)  

22. Example using table Find (a) lim ┬(𝑥→2^− )⁡𝑓(𝑥) (b) lim ┬(𝑥→2^+ )⁡𝑓(𝑥) 1.9 3.9 2.1 4.1 1.99 3.99 2.01 4.01 1.999 3.999 2.001 4.001 1.9999 3.9999 2.0001 .0001 1.9 3.9 2.1 4.1 1.99 3.99 2.01 4.01 1.999 3.999 2.001 4.001 1.9999 3.9999 2.0001 .0001

23. Example cont. Find (a) (b)  

24. Class Activity: Find the limit using the graph of functions.

25. Graphically:

26. Using Table:  

27. Explanation: = 3 (From the table, we noted that x gets closer to 2 from the left then the values of f approaches 3) ( From the table, we noted that x gets closer to 2 from the right then the values of f approaches 0) The limit from the left does not equal the limit from the right. So, the Limit does not exists  

28. Theorem if, and only if,  

29. Class Activity: What is limit from the left at x=1? (b) What is limit from the right at x=1? (c) Does limit exists at x=1? Explain.

30. Solution:

31. Example: The graph of a function is given. State the Value of each quantity, if it exists. If it does not exist, explain why not.  

32. 1.3 The Limit Laws Basic Limits: For any real number and any constant   The limit of identity function is equal to the number is approaching. The limit of a constant function is equal to the constant. The limit of a constant function is equal to the constant.

33. Basic limits: Explanation

34. Summary of Limit Laws

35. Examples

36. Example Use the limit laws to evaluate Solution : Let’s apply the limit laws one step at a time to be sure we understand how they work. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied. Apply the difference law Apply the constant multiple law Apply the basic limit result and simplify  

37. Example Use the limit laws to evaluate Solution: To find this limit, we need to apply limit laws several times. Again, we need to keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to be applied. Ensure that limit of the denominator is not equal to zero.  

38. Example

39. Example cont.

40. Example Use the limit laws to evaluate In each step, indicate the limit law applied. Solution : Apply Product law for limits, Apply the constant multiple law  

41. Limit of Polynomial functions Let be polynomial functions. Let be a real number. Then, Explanation: The limit of a polynomial as approaches is the .  

42. Example: Evaluate a limit of a polynomial Solution Let Explanation: Note that is a polynomial and since where .  

43. Limits of Rational Functions: Let and be polynomial functions. Let be a real number. Then, Explanation: Provided the limit of rational function is a function value at When , is said to be in the domain of the rational function .  

44. Example Evaluating a Limit of a Rational Function

45. Problem-Solving Strategy: Calculating a Limit When f ( x )/ g ( x ) has the Indeterminate Form 0/0

46. Evaluating a Limit by Factoring and Canceling Example

47. Example cont.

48. Evaluating a Limit by Multiplying by a conjugate

49. Example cont.

50. Evaluating a Limit when the Limit Laws Do Not Apply

51. Example cont.

52. Class Activity Evaluate the limit Cancellation (b) Conjugate  

53. Evaluating a Two-sided Limit Using the Limit Laws

54. Solution

55. Infinite limits from the left Let be a function defined at all values in an open interval of the form ( i ) If the values of increase without bound as the values of (where ) approach the number a, then we say that the limit as approaches a from the left is positive infinity and we write (ii) If the values of decrease without bound as the values of (where ) approach the number , then we say that the limit as approaches a from the left is negative infinity and we write  

56. Infinite Limits from right Let be a function defined at all values in an open interval of the form If the values of increase without bound as the values of (where ) approach the number a, then we say that the limit as x approaches a from the left is positive infinity and we write ii. If the values of decrease without bound as the values of (where approach the number a, then we say that the limit as approaches a from the left is negative infinity and we write  

57. Two-sided infinite Limit Let be defined for all in an open interval containing a ( i )If the values of increase without bound as the values of (where ) approach the number a, then we say that the limit as approaches is positive infinity and we write (ii) If the values of f(x) decrease without bound as the values of (where approach the number , then we say that the limit as x approaches a is negative infinity and we write  

58. Infinite Limits (a)   Explanation: From the graph of the function we see that as x approaches zero from the left decrease without bound. so, as x approaches zero from the right increase without bound. so,  

59. Two-sided Limit   Explanation: From the graph of the function we see that as x approaches 1 from the left increase without bound. We have as x approaches 1 from the right increase without bound. We have We see that f(x) increases without bound as x approaches 1. Hence we have  

60. Vertical Asymptote

61. Example of vertical asymptote (a)   Since The line x =0 is a vertical asymptote. Since The line x =0 is a vertical asymptote.  

62. Conti. (b) For   We have seen from above So, the line x =0 is a vertical asymptote.  

63. Finding a Vertical Asymptote

64. Behavior of a function at Different Points Example: Use the graph below to determine each of the following

65. Limits at infinity If the values of f(x) become arbitrarily close to L as x becomes sufficiently large, we say the function f has a limit at infinity and write If the values of f(x) becomes arbitrarily close to L for x<0 as |x| becomes sufficiently large, we say that the function f has a limit at negative infinity and write If the values f(x) are getting arbitrarily close to some finite value L as x→ or x→− , the graph of f approaches the line y=L. In that case, the line y=L is a horizontal asymptote of f .  

66. Horizontal Asymptote If or , we say the line y=L is a horizontal asymptote of f.  

67. Explanation: (a) As x→ , the values of f are getting arbitrarily close to L. The line y=L is a horizontal asymptote of f (b) As x→− , the values of f are getting arbitrarily close to M. The line y=M is a horizontal asymptote of f. IMORTANT: A function cannot cross a vertical asymptote because the graph must approach infinity (or − ) from at least one direction as x approaches the vertical asymptote. However, a function may cross a horizontal asymptote.  

68. Example: Explanation: The function approaches the asymptote y=2 as x approaches Values of a function f as x :  

69. Example For each of the following functions f, Evaluate Determine the horizontal asymptote(s) for f.  

70. Squeeze Theorem

71. Graphical representation of Squeeze Theorem Explanation: The Squeeze Theorem applies when and  

72. Applying Squeeze theorem: Apply the Squeeze theorem to evaluate Solution: Because , for all x, we have Since from the Squeeze theorem. We obtain The graphs of are shown in graph. The graphs of and Are shown around the point x=0.  

73. Class Activity Use the Squeeze theorem to evaluate  

74. Determine Limits Numerically: Video Examples Ex 1:  Determine a Limit Numerically Ex 2:  Determine a Limit Numerically Ex 3:  Determine a Limit Numerically

75. Determine Limits Graphically: Video Examples Examples:  Determining Basic Limits Graphically Ex 1: Determine Limits from a Given Graph Ex 2: Determine Limits from a Given Graph

76. Determine One-Sided Limits Graphically Ex 1:  Determining Limits and One-Sided Limits Graphically Ex 2:  Determining Limits and One-Sided Limits Graphically

77. Determining Limits Analytically Ex 1:  Determine a Limit Analytically Ex 2:  Determine a Limit of a Piece-Wise Defined Function Analytically Ex:  Determine Limits of a Piecewise Defined Function Ex: Determining Limits of Rational Functions by Factoring Ex 2: Determine a Limit of a Rational Function by Factoring and Simplifying

78. Infinite Limits: Videos Determine Infinite Limits of a Rational Function Using a Table and Graph

79. Limits at Infinity Limits at Infinity Determine Limits at Infinity Numerically Using a Desmos Ex:  Determining Limits at Infinity Graphically

80. Limits at Infinity and Horizontal Asymptotes Determine Limits at Infinity and Equations of Horizontal Asymptotes from a Graph Determine Limits and Equations of Asymptotes from a Graph (Rational) Determine Limits at Infinity of Rational Functions Using 2 Methods: Degree and Dividing Determine Limits at Infinity of Rational Functions Using Highest Degree Terms Limit at Infinity of a Rational Function Using 2 Methods: Degree and Dividing (Constant) Limit at Infinity of a Rational Function Using 2 Methods: Degree and Dividing: (Infinity) Limits at Infinity of a Rational Function Using 2 Methods: Degree and Dividing: (Zero)