Chapter 10: Limit of a Function. SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M.

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1 Chapter 10: Limit of a Function SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

2 Chapter 10: Limit of a Function Lecture 10.1: Limit of a Function Lecture 10.2: One-Sided Limits Lecture 10.3: Limit Theorems Lecture 10.4: Limits of Transcendental Functions

3 Lecture 10.1: Limit of a Function SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

4 Something to think about As for your daily concerns, how do you usually eperience limiting some of your personal activities?

5 Physical Limit: Monitoring our speed limit in an epressway.

6 Physical Limit: Being health conscious by limiting the salt and sugar consumption.

7 Physical Limit: Adjusting the weight capacity of a bridge done by engineers.

8 Limit A limit is something that can or cannot be reached but can possibly be eceeded.

9 Geometric Application of Limits:

10 Our Observation: As the number of sides of the polygon increase, the polygon is getting closer and closer to becoming the circle!

11 Application of Limits on Sequence: a n n 1 n 1 2, 2 3, 3 4, 4 5, 5 6,..., 10 11,..., ,...,

12 Our Observation: As n gets bigger and bigger, n/(n+1) gets closer and closer to 1.

13 Application of Limits on Sequence: a n 1 1 n 1, 1 2, 1 3, 1 4, 1 5,..., 1 100,..., ,...,

14 Our Observation: As n gets bigger and bigger, 1/n gets closer and closer to 0.

15 Eample : Let us discuss the behavior of a function close to a certain point. For eample, f ( ) 2 as approaches 2.

16 Solution: F()

17 Graph :

18 Final Answer: This value 4 is said to be the limit, or limiting value, of the function when approaches the value 2, and we write this as: lim 2 f ( ) 4 or lim

19 Limit Let f() be a given function of. If the function f() approaches the real number L as approaches a particular value of c, then we say that L is the limit of f as approaches c. The notation for this definition is: The limit lim c lim c f ( ) L f ( ) L may or may not eist (as we will see later). When the limit f() eist, it must be a single number.

20 Eample : The function f ( ) 2 9 is not 3 defined when = 3. Why? What happens to the values of f() when is very close to 3?

21 Solution: F()

22 Graph :

23 The Hole or Open Circle The hole or open circle in the previously given figure represents a break in the graph, indicating that f() is not defined when = 3.

24 Final Answer: Thus, we conjecture that the limit is 6, and we have 2 9 lim

25 Take Note: lim f ( ) is NOT f(c) c lim ( ) It must be emphasized that c is different from f(c). The former, if it eists, gives the value that f() approaches as gets closer to c. It is not about what f() becomes when is c. That would be f(c). f

26 Eample : If f ( ) 4, what is 2 lim 4 4 2?

27 Solution: F()

28 Graph :

29 Final Answer: The table suggests that as approaches 4 from either direction, f() gets closer and closer to 4 and so we have lim 4 f ( ) 4, 4 lim or equivalently,

30 Take Note: The limit of a function f() may eist as approaches c even if f() is not defined at = c.

31 Eample : Use numerical evidence to make a conjecture about the value of lim 0 sin.

32 Solution: (radi ans) ±1.0 ±0.5 ±0.4 ±0.3 ±0.2 ±0.1 ±0.01 ±0.001 F()

33 Graph :

34 Take Note: Take note that both positive and negative values of give us the same value for f.

35 Final Answer: We observe that as approaches 0 from the left or the right, f() approaches 1. That is, lim 0 sin 1.

36 Classroom Task 10.1: Please answer "Let's Practice (LP)" Number 37.

37 Lecture 10.2: One-Sided Limits SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

38 Recall: The notation c means that approaches c. It requires to be on either side of c; that is, is either less than c or greater than c.

39 approaches c from the right or from above It may be that we wish to have approach c but we want values close to c which are always larger than c. If this is the case, we say that approaches c from the right or from above, and we use the notation: c

40 approaches c from the left or from below If approaches c but always stays less than c, we say that approaches c from the left or from below, and we use the notation: c

41 One-Sided Limits If f is a function defined near c, the c limits of f() as or are called one-sided limits of f. The righthand limit of f at c is denoted by lim c f ( ), and the left-hand limit of f at c is denoted by c lim c f ( ).

42 Formal Definition of One-Sided Limits For any function f, that, as approaches c, f() means approaches L. Alternatively, if lim f ( ) L and lim f ( ) L, c then lim c f ( ) c lim c f L ( ) L.

43 Remark 1: If the left-hand limit does not equal the right-hand limit, then there is no limit. In this case, we say that the limit does not eist.

44 Figure :

45 Remark 2: The eistence lim c f ( ) of does not depend on whether f(c) is defined or not.

46 Figure :

47 Remark 3: The eistence of lim c f ( ) does not depend on the value of f(c) if f(c) is defined. That is f(c) is defined, but does not equal lim f ( ). c

48 Figure :

49 Eample : Determine the onesided limits of f as 0. ( )

50 Graph :

51 Final Answer: Thus, as approaches 0 from both sides, the corresponding value of f() do not get closer and closer to a single real number. Therefore, the LIMIT DOES NOT EXIST.

52 Eample : For the functions in the following figures, find the one-sided and two-sided limits at = c if they eist.

53 Figure :

54 Figure :

55 Figure :

56 Eample : Find lim

57 Solution: F()?

58 Final Answer: That is, 2 1 lim

59 Limits of Piecewise- Defined Functions One-Sided Limits SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

60 Piecewise-Defined Functions In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain (a sub-domain).

61 Eample : Estimate lim 2 f ( ) for f ( ) 1, 4, 2 2.

62 Solution: f()

63 Solution: f()

64 Graph :

65 Our Conclusion: We conclude that: lim 2 f ( ) 1 whereas: lim f ( ) 4. 2

66 Final Answer: Since the left-hand limit does not equal the right-hand limit, we conclude that lim 2 f ( ) DOES NOT EXIST.

67 Classroom Task 10.2: Please answer "Let's Practice (LP)" Number 38.

68 Lecture 10.3: Limit Theorems SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

69 Property 1: Limit of a Constant Function lim c k k for any constant k That is, the limit of a constant function is the constant.

70 Eample : lim 1 4 4

71 Property 2: Limit of a Polynomial Function If f() is a polynomial, lim c f ( ) ( c). This means that if you are looking for the limit of a polynomial function as approaches c, you can just evaluate the function at = c to find the limit. f

72 Eample : 4? lim 1

73 Illustration: lim 1

74 Properties of Limits lim c Let f and g be two functions, and assume that: f ( ) L and lim c g( ) M where L and M are real numbers (both limits eist). Then:

75 Property 3: Limit of the Sum That is, the limit of the sum is the sum of limits.. ) ( lim ) ( lim ) ( ) ( lim M L g f g f c c c

76 Property 4: Limit of the Difference That is, the limit of the difference is the difference of limits.. ) ( lim ) ( lim ) ( ) ( lim M L g f g f c c c

77 Property 5: Limit of a Constant Times a Function lim c f ( ) c lim c L c c This says that the limit of a constant times a function is the constant times the limit of the function.

78 Eample : 3 2? lim 2 1

79 Illustration: lim (1)

80 Property 6: Limit of the Product In words, the limit of the product is the product of the limits. M L g f g f c c c ) ( lim ) ( lim ) ( ) ( lim

81 Eample : lim ?

82 Illustration: lim (11)(2) 22

83 Property 7: Limit of the Quotient f ( ) lim f ( ) lim c c g( ) lim g( ) c M The limit of the quotient is the quotient of the limits. L

84 Eample : 2 lim 8? 2

85 Illustration: lim

86 Property 8: Limit of a Function Raised to an Eponent lim c n n f ( ) lim f ( ) L n c The limit of a function raised to an eponent is the limit of the function raised to that eponent.

87 Eample : 2 3? lim 4

88 Illustration: lim 4

89 Property 9: Limit of the nth Root of a Function lim c n f ( ) n lim c f ( ) n L The limit of the nth root of a function is the nth root of the limit of the function. However, when n is even, we must have L 0.

90 Eample : lim 2 7?

91 Illustration: lim 2

92 Eample : Find lim

93 Final Answer: lim , 807

94 Eample : Find lim 2 1

95 Final Answer: lim 2 1

96 Eample : 2 lim 2 Find

97 Final Answer: 2 lim

98 Eample : Find lim

99 Final Answer: lim

100 Classroom Task 10.3: Please answer "Let's Practice (LP)" Number 39.

101 Lecture 10.4: Limits of Transcendental Functions SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza

102 Limits of Transcendental Functions So far, we have been eamining the limits of algebraic functions. In this section, we focus our attention to some transcendental (trigonometric, eponential, logarithmic) functions.

103 Eample : Find sin t lim t 0

104 Solution: F()

105 Final Answer: Thus, lim 0 sin t 0. t

106 Eample : Find sin t lim t 0 t

107 Solution: F()

108 Final Answer: The table suggests that lim t 0 sin t t 1.

109 Eample : Now, we investigate 1 cos t lim 0 t t

110 Solution: F()

111 Final Answer: Here, the table suggests that 1 cos t lim 0 0. t t

112 Eample : Let us now look at the behavior of the eponential function lim 1 e

113 Solution: F()

114 Final Answer: This suggests that lim e

115 Eample : Net, let us look at the behavior of the logarithmic function lim 1 ln

116 Solution: F()

117 Eample : This suggest that. lim1 ln 0

118 Classroom Task 10.4: Please answer "Let's Practice (LP)" Number 40.