PreCalculusChapter4Limits.........................

PRECAL LIMITS

1. Chapter 4 (Limits)

2. Concept of a Limit Consider a function defined by whose domain is the set of all real numbers except 1, since is not defined when    

3. : Concept of a Limit Illustration 1 A case where the limit of as is not . Suppose a function is modified in this manner   Note: the existence of a limit of a function as (for one side or both sides) does not equal in whether is defined at but only on whether is defined for near the number  

4. : Concept of a Limit Illustration 2 A limit that does not exist   0.9 21.9 0.99 201.99 0.999 2001.999 0.9999 20001.9999 0.9 21.9 0.99 201.99 0.999 2001.999 0.9999 20001.9999 1.1 -197 1.01 -197.99 1.001 -1997.999 1.0001 -19997.9999 1.00001 -199997.9999 1.1 -197 1.01 -197.99 1.001 -1997.999 1.0001 -19997.9999 1.00001 -199997.9999 does not exist   does not exist   Therefore, Does not exist  

5. : Concept of a Limit Definition: Two-sided Limits A function that is defined at some open interval has the limit as , written If we can get as close to as we wish by taking sufficiently close to , but not equal to  

6. : Concept of a Limit Definition: One-sided Limits (Left hand) If a function can be made arbitrary close to a number by taking sufficiently close to, but not equal to a number from the left (Right hand) If a function can be made arbitrary close to a number by taking sufficiently close to, but not equal to a number from the right  

7. : Concept of a Limit Illustration 3: A limit that exists on one side Domain:   does not exist  

8. : Concept of a Limit Determine the following limits        

9. : Concept of a Limit Indeterminate Form If the limit of a function of the form as results to , then we say that the function is in indeterminate form which means that a limit may or may not exist depending on the situation   Other indeterminate forms (photo credit: @ fermatslibrary )

10. : Concept of a Limit Indeterminate Form        

11. : Concept of a Limit Formal Definition of a Limit -- A function defined at some open interval , then the limit as is written as *if for every there is a corresponding number such that whenever  

12. : Concept of a Limit Consider   Intuitively, we find that , because is close to 5 as is close to 2  

13. : Concept of a Limit Say we want to prove that   By particularizing on this side: -- if f(x) can be made arbitrarily close to 5 by taking x sufficiently close to 2, from either side but different from 2, then , then we need to make the concepts of arbitrarily close and sufficiently close to precisely say we want the distance between the following number and 5, use less than 0.1, that is            

14. : Concept of a Limit Say we want to prove that     Subtracting by   Using absolute value and remembering that   Thus for an arbitrarily close to 5 of .1 sufficiently close to 2 means within .05   Then the distance of from 5 is guaranteed to satisfy  

15. : Concept of a Limit Say we want to prove that   Another way of expressing it is when is a number different from 2 between in the open interval (1.95 , 2.05) on the x-axis and is on the interval (4.9 , 5.1) on the y-axis.                 If we get whenever  

16. : Concept of a Limit 2. Prove Use   Proof: -- For any arbitrary regardless how small we wish to find so that whenever   whenever   whenever   whenever   whenever   whenever     Verify:   -axis:   -axis:  

17. : Concept of a Limit 3. Let the function to be defined by , given find a for   whenever   whenever   whenever   whenever   whenever    

18. Try this! : Concept of a Limit 4. Prove  

19. Theorems on Limits Limit Theorem 1 (Limit of a Linear Function) -- if is a linear function, then or   1

20. 1 Proof:   We must show that for every , there exists a such that   We want to find a , for any such that whenever or since , when   Hence, to make this statement hold, we choose so we conclude that if and , then equivalently whenever   Illustration:  

21. Limit Theorem 2 (Constant functions) -- if is a constant, then for any number   The hint is that, when the approaching variable is absent on the function, then the entire function is a constant function. Same idea applies with different letters aside from the approaching variable (theorem 4). Examples:   Illustration   2

22. Limit Theorem 3 (Limit of Identity Functions)   This follows immediately for the limit   Limit Theorem 4 (Limit of Constant Multiple Functions) -- if is a constant, then for any number   3, 4 Illustration Now  

23. 5 Limit Theorem 5 (Addition Rule of Limits) If and , then       We must prove that there exists a such that whenever   Since whenever whenever   Now let be the smaller of the two and , and so we can say that   whenever   whenever  

24. 5 Limit Theorem 5 (Addition Rule of Limits) If and , then       whenever   whenever   Hence whenever  

25. Limit Theorem 6 (Product Rule of Limits) If and , then       Limit Theorem 7 (Quotient Rule of Limits) If and , then       6,7 Evaluate:      

26. 8 Limit Theorem 8 (Power Rule of Limits) -- Suppose is a positive energy and , thus       Proof: (Using Theorem 6)       Corollary: -- if is appositive integer, then     Corollary Proof: (Using Theorem 8) *Suppose    

27. 9 Limit Theorem 9 (Limits of Polynomial Functions) -- For any polynomial , then for any number   Proof: Consider a polynomial function such that then     Corollary: -- if and are polynomial functions, then    

28. 1, 2, 3, 4, 5, 6, 7, 8, 9                           *Tip: when evaluating an algebraic expression with a variable set to 1, simply remove the variable and evaluate the remaining expression      

29. 1, 2, 3, 4, 5, 6, 7, 8, 9   When substituting , the function will go indeterminate, if this happens, the limit may or may not exist, but it is necessary to manipulate the equation first                        

30. 10 Limit Theorem 10 -- Suppose is a positive integer and , then *provided that when is even     Corollary: -- If is a function such that where is a polynomial, then -- If is a positive integer, then *provided that where is constant   Corollary: -- If and are polynomial functions and then  

31. 10 Limit Theorem 10 -- Suppose is a positive integer and , then *provided that when is even     Corollary: -- If and , then the following limit does not exist  

32. 10   Corollary: -- If and , then the following limit does not exist   Proof: -- Suppose and , and suppose further that exists and is equal to       *note that this expression means the same since multiplying and dividing by the same quantity does nothing       This is called proof by contradiction. By contradicting the initial assumption , we have proven the above statement.  

33. Limit Theorem 11 (Uniqueness of a Limit) -- If , the is unique     10, 11                    

34. 10, 11     If the function involving radicals becomes indeterminate or undefined, we should not declare that the limit does not exist right away. We must multiply both the numerator and the denominator with the conjugate of the radical term                        

35.   10, 11 Note that the conjugate is the term needed to be multiplied for the expression to be the difference between two powers of (where ) Those are: So in this case, the conjugate is the term needed in order for it to become difference of two cubes            

36.   1, 2, 3, 4, 5, 6, 7, 8, 9, 10        

37. Heaviside Function -- Given a function defined by   Evaluate:  

38. Greatest Integer Function -- Given a function defined by if such that -- The symbol is defined to be the greatest integer less than or equal to       *Note that it follows that for  

39.   Greatest Integer Function -- Given a function defined by if such that -- The symbol is defined to be the greatest integer less than or equal to  

40.   In evaluating limits of two variables, it is necessary to evaluate separate limits for every variable, but keeping the values correspondant .    

41.   2, 4 As you might have noticed before, the number(s) on the lower left are the limit theorem (number/s) present on a slide are the limit theorem/s defined at that slide and/or the theorem(s) that can be used as the concept for the example in that slide. Where are constants          

42. Try these!             Note: If the limit does not exist, state “the limit does not exist”.

43. Limits of Elementary Transcendental Functions Transcendental Functions are functions that are not algebraic in nature Trigonometric Inverse Trigonometric Exponential Logarithmic Hyperbolic Trigonometry Inverse Hyperbolic Trigonometry *Before starting with the trigonometric functions, there will be a few slides (after this one) dedicated to recall the different trigonometric identities

44. Trigonometric Magic Hexagon -- Used to identify relationships between the six trigonometric functions This is the structure of the hexagon, in order to be familiarized with it, here are some keypoints : -- all functions starting with c are at the right side, and are in alphabetical order -- sin and sec are at the upper left and lower left respectively -- tan is at the center left -- there is a “1” at the center Notes: -- and might be interchangeably used with the functions, they mean the argument of the function -- limited examples were given, not everything. However, it was provided how they work  

45. Multiplication Property -- the product of two functions is the function in between them (five examples, one click per example) : Trigonometric Magic Hexagon         We can also make use of the “1” in the center   There are still many multiplication pairs out there, but that is how that goes. *If two functions don’t have a function in between them, then they are multiplied like this:  

46. : Trigonometric Magic Hexagon Division Property -- involves three functions and rotating whether clockwise or counter-clockwise. The first function is equal to the second function over the third (5 examples)          

47. : Trigonometric Magic Hexagon Reciprocal Property -- a corollary of the division property is the reciprocal property wherein rather than by rotating, we go through the “1” at the center (3 pairs of examples)      

48. : Trigonometric Magic Hexagon Complementary Property -- the complement of a function is its counterpart on the opposite side (3 pairs of examples)      

49. : Trigonometric Magic Hexagon Pythagorean Identities -- is achieved by rotating clockwise inside an inverted triangle (first plus the second equals the third, all terms are squared)       *From there, you can derive the other forms such as  

50. Several identities not covered by the hexagon are found here and the next 4 slides. Sum and Difference of Two Angles   or ? *the signs accord with what goes on top and bottom, in the cosine one, if the sign used on the left side is +, we will use – on the right side, and vice-versa. For the tangent, if the sign on the left side is +, the denominator will use the -. *for those that have both on one side, if the left side is +, then the right side is +, and vice-versa  

51. Double Angle Identities     * has three possible values   Half-Angle Identities     * has three possible values   Triple Angle Identities  

52. Powers of Functions   Product of Functions   Sum and Difference of Functions  

53. Degree Equivalent - Degree Equivalent : Trigonometric Functions for Special Angles (blank spaces mean undefined)

54. The Unit Circle (An ordered pair is )   Photo credit: Jim Belk

55. Limit of Trigonometric Functions Consider the limit: (for convenience, this presentation will use for radians and for degrees, unless otherwise specified)   Substituting will yield to a situation; however   -.1 0.998334166 -.01 0.999983333 -.001 0.999999833 -.0001 0.999999998 ? .0001 0.999999998 .001 0.999999833 .01 0.999983333 .1 0.998334166 -.1 0.998334166 -.01 0.999983333 -.001 0.999999833 -.0001 0.999999998 ? .0001 0.999999998 .001 0.999999833 .01 0.999983333 .1 0.998334166  

56. Limit Theorem 12 -- For any real number   Limit Theorem 13 -- For any real number , given that these exists   12, 13

57. 12, 13                

58. 12, 13     Note that , but there is no in the angles of the unit circle. However, note that when going counter-clockwise, the angle is positive; and going clockwise, the angle is negative. Therefore, we will go clockwise radians. We will land on  

59. 12, 13       By remembering that an ordered pair is          

60. 12, 13   Substituting will yield to an indeterminate situation. However, from the Pythagorean Identity We can derive for                 Another indeterminate form. But since we know that             *Surely, those are terrible ways to express “2”

61. 12, 13   Although the numerator will not yield zero to threaten for an indeterminate form, every term themselves in the denominator are undefined, so some work still has to be done. From the Pythagorean identities we can derive that From we can derive that           Trigonometric functions have conjugates too.              

62.   We can go on say that from Pythagorean identity then, we will have     But (if you still remember that hexagon)     12, 13      

63. 12, 13   We can factor the numerator, such that                          

64. 12, 13           There is nothing wrong in trying to substitute directly the value first. If it failed, then, we could go on working on with the function.      

65. 14 Squeeze Theorem (Pinching or Sandwiching Theorem) -- This theorem states that the graph of is squeezed between the graphs of two other functions and for all close to and if the functions and have a common limit as approaches , it stands to receive that also approaches as approaches .   L imit Theorem 14 -- Suppose that , , and are functions, defined on some interval , containing except possibly at itself , for which Also suppose that then  

66. 14 Proof: -- Given , we can choose to get within of We can also choose to get within of . If , then whenever , we also have Similar arguments prove that the squeeze theorem hols for limits as , and and  

67. 14 Illustrate: Let the function , , and be defined by         Since  

68. 14 Illustrate:     Since  

69. Illustrate: 14    

70. 14 13.) Use squeeze theorem to prove that   Since only, then we can say that the boundary for the values of is   For all if , then Multiplying all sides by gives us              

71. 14       For those who want some satisfaction behind the name of the theorem, here are the graphs from afar and close-up. The functions are color-coded From afar If we zoom it in to where it compresses, you can see that bounces up and down in between and as if it is “squeezed” between those two functions  

72. 14            

73. Limit Theorem 15 *where is the argument of sine   15

74. 15 Proof: Consider a sector of a unit circle, where and are the radii with measure 1.  

75. 15 Proof: Observe that a triangle is formed when we extend the segment and let the intersection when joins the extended   Let , then   Kindly take note of the figures and values as we will put them together later  

76. 15 Proof: Form and let be its height, then   Kindly take note of the figures and values as we will put them together later  

77. 15 Area of   Area of sector   Area of                    

78. 15 Multiply all by , and don’t forget that we have an inequality         (reciprocate)   (mirror) Notice:    

79. 15 Illustration: the cosine function is green, y=1 is red, and is blue  

80. 14, 15                            

81. 16 Limit Theorem 16 *where is the argument of cosine   Proof:                   Same process follows for the other one

82. 14, 15, 16   Try this example first. But note that this is not the proper way of doing it.       Note that the 2 and 7 will not have any effect on the function since they will just cancel each other. The same goes with the over x, since they are both in the numerator and the denominator      

83. 14, 15, 16                   We can do the conjugate of the numerator, but if someone has noticed, the function simplifies to be this by half-angle identity (refer to slide 51)      

84. 14, 15, 16   Unlike some major examples in the other chapters, this humongous problem did not came from any math contests. Again, do not be intimidated by large problems (especially this one that involves trigonometry), and try this problem first. The solution will be on the next click.        

85. Try these!           Note: If the limit does not exist, state “the limit does not exist”.

86. B: Limits of Inverse Trigonometric Functions Let   -.1 -0.1001674212 -.01 -0.01000016667 -.001 -0.001000000167 -.0001 -0.0001000000167 .0001 0.0001000000167 .001 -0.001000000167 .01 0.01000016667 .1 0.1001674212 -.1 -0.1001674212 -.01 -0.01000016667 -.001 -0.001000000167 -.0001 -0.0001000000167 .0001 0.0001000000167 .001 -0.001000000167 .01 0.01000016667 .1 0.1001674212  

87. Limit Theorem 17 -- For any number *such that and   Limit Theorem 18 -- For any number *such that and   Limit Theorem 19 -- For any number [ 2] *such that   [2] Can be written as   17, 18, 19

88. Illustration: Graph of inverse sine , inverse cosine , inverse tangent , inverse cotangent , inverse secant , inverse cosecant in domain   17, 18, 19

89. 17, 18, 19     But there can be more than one value for inverse sine and inverse cosine of And those were only positive values!   But we just have to stick with the values for these inverse functions as stated in the previous slides, so we have     Note: You can access the inverse trigonometric functions for sine, cosine, and tangent in your calculator when you press shift, then the function: shift+sin =sin -1 shift+cos =cos -1 shift+tan =tan -1

90. 17, 18, 19                        

91.   Note:   18   But is and on the first revolution, neither of which is in the interval   Since , then the value we are looking for is between quadrant IV and quadrant I   By rotating radians clockwise, we arrive back to , which is the same angle.     Put this slide on slideshow view for some visualization not visible in plain sight

92.   Note:   18 Put this slide on slideshow view for some visualization not visible in plain sight   Values given are on the first revolution     is in the range of inverse cotangent, so we can take that, but the two values of inverse cotangent are not, therefore, we will do rotating counter-clockwise again      

93.   Note:   18 Values given are on the first revolution          

94.   Note:   18 Values given are on the first revolution        

95.   Note:   18 Values given are on the first revolution Put this slide on slideshow view for some visualization not visible in plain sight      

96. Try these! 17, 18, 19     This powerpoint file already has 96 slides and has more than 4.69MB file size, to avoid further problems with the memory, the next parts of this chapter will be on another file.