Chapter 2: Functions, Limits and Continuity

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1 Chapter 2: Functions, Limits and Continuity Functions Limits Continuity Chapter 2: Functions, Limits and Continuity 1

2 Functions Functions are the major tools for describing the real world in mathematical terms. Definition 1. A function f is a set of ordered pairs of numbers (x, y) satisfying the property that if (x, y 1 ), (x, y 2 ) f then y 1 = y 2 (that is, no two distinct ordered pairs in f have the same first component). If f is a function from a set X into a set Y, then we write f : X Y read as f is a function from set X into set Y. If (x, y) f, then we may write y = f(x) (read as y equals f of x ) Chapter 2: Functions, Limits and Continuity 2

3 so that y is called the image of x under the function f; x is a pre-image of y under f. We also say that y is the function value of x under f. In the function f : X Y, the set X containing all of the first components of ordered pairs in f is called the domain of the function f; the set Y is called the co-domain of f. The set of all second components of ordered pairs in f is called that range of f, that is, range f = {y Y : y = f(x) for some x X}. Clearly, range f Y. Chapter 2: Functions, Limits and Continuity 3

4 Example 1. Let f = {(x, y) : y = x 2}. The value of y that corresponds to x = 6 is y = 6 2 = 4 = 2. Hence, f(6) = 2 and (6, 2) f. The domain of f is the set and the range of f is the set dom f = [2, + ) range f = R 0 = [0, ). Chapter 2: Functions, Limits and Continuity 4

5 Definition 2. If f is a function, then the graph of f is the set of all points (x, y) in a given plane for which (x, y) f. Example 2. The graph of f(x) = x 2 is given by Chapter 2: Functions, Limits and Continuity 5

6 Example 3. The graph of g(x) = x 2 4 is given by Chapter 2: Functions, Limits and Continuity 6

7 Example 4. The graph of h(x) = x 2 9 is given by Chapter 2: Functions, Limits and Continuity 7

8 Example 5. The graph of h(x) = x(x 2) is given by Chapter 2: Functions, Limits and Continuity 8

9 Example 6. The graph of F (x) = 3x 1, if x < 2 4, if x = 2 7 x, if x > 2. is given by Chapter 2: Functions, Limits and Continuity 9

10 Example 7. The graph of G(x) = { 4 x 2, if x 2 10, otherwise. is given by Vertical Line Test A vertical line intersects the graph of a function f in at most one point. Chapter 2: Functions, Limits and Continuity 10

11 Limits Example 8. How does the function f(x) = 2x2 + x 3 x 1 behave near x = 1? x f(x) = 2x2 + x 3 x x less than 1 x f(x) = 2x2 + x 3 x x greater than 1 Chapter 2: Functions, Limits and Continuity 11

12 Definition 3. (Informal Definition of Limit) Let f(x) be defined on an open interval containing a, except possibly at a itself. If f(x) gets arbitrarily close to L for all x sufficiently close to a, we say that f approaches the limit L as x approaches a, and we write lim f(x) = L. x a Example 9. Let f be defined by f(x) = x 2 1 x 1, if x 1 1, if x = 1 Chapter 2: Functions, Limits and Continuity 12

13 Example 10. Find the following limits (if they exist). 1. lim x lim x 2 x 3. lim x 0 (5x 3) Chapter 2: Functions, Limits and Continuity 13

14 Remark 1. A function may fail to have a limit at a point in its domain. Example 11. Discuss the behavior of the following functions as x approaches a = U(x) = { 0, x < 0 1, x 0 Chapter 2: Functions, Limits and Continuity 14

15 2. g(x) = { 1/x, x 0 0, x = 0 Chapter 2: Functions, Limits and Continuity 15

16 3. f(x) = { 0, x 0 sin(1/x), x > 0 Chapter 2: Functions, Limits and Continuity 16

17 Limits from Graphs For the given function f(x) with its graph shown, find the following limits. Example lim x 1 f(x) 2. lim x 2 f(x) 3. lim x 3 f(x) Chapter 2: Functions, Limits and Continuity 17

18 Example lim f(x) x 2 2. lim x 1 f(x) 3. lim x 0 f(x) Chapter 2: Functions, Limits and Continuity 18

19 Example lim x 2 f(x) 2. lim x 1 f(x) 3. lim x 0 f(x) Chapter 2: Functions, Limits and Continuity 19

20 Definition 4. (Formal Definition of Limit) Let f be a function defined at every number in some open interval containing a, except possibly at the number a itself. The limit of f(x) as x approaches a is L, written as lim f(x) = L x a if the following statement is true: Given any ɛ > 0, however small, there exists a δ > 0 such that if 0 < x a < δ then f(x) L < ɛ. Chapter 2: Functions, Limits and Continuity 20

21 Example 15. Let f(x) = 2x Find a δ > 0 such that whenever 0 < x 3 < δ whenever f(x) 1 < ɛ where ɛ = Show that lim x 3 f(x) = 1. Chapter 2: Functions, Limits and Continuity 21

22 Example 16. Let f(x) = x Find a δ > 0 such that whenever 0 < x 2 < δ whenever f(x) 4 < ɛ where ɛ = Show that lim x 2 f(x) = 4. Chapter 2: Functions, Limits and Continuity 22

23 The Limit Theorems Theorem 1. (Limit of a Linear Function) If m and b are constants, then lim mx + b = ma + b. x a Theorem 2. (Limit of a Constant) If c is a constant, then for any number a lim c = c. x a Theorem 3. (Limit of the Identity Function) lim x = a. x a Chapter 2: Functions, Limits and Continuity 23

24 Theorem 4. (Limit of the Sum and Difference of Two Functions) If lim f(x) = L and lim g(x) = M, then x a x a lim [f(x) ± g(x)] = L ± M. x a Theorem 5. (Limit of the Sum and Difference of n Functions) If lim f 1 (x) = L 1, lim f 2 (x) = L 2,..., lim f n (x) = L n, then x a x a x a lim [f 1(x) ± f 2 (x) ± ± f n (x)] = L 1 ± L 2 ± ± L n. x a Theorem 6. (Limit of the Product of Two Functions) If lim f(x) = L and lim g(x) = M, then x a x a lim [f(x)g(x)] = LM. x a Chapter 2: Functions, Limits and Continuity 24

25 Theorem 7. (Limit of the Product of n Functions) If lim f 1 (x) = L 1, lim f 2 (x) = L 2,..., lim f n (x) = L n, then x a x a x a lim [f 1(x) f 2 (x) f n (x)] = L 1 L 2 L n. x a Theorem 8. (Limit of the nth Power of a Function) If lim x a f(x) = L and n is any positive integer, then lim x a [f(x)]n = L n. Theorem 9. (Limit of the Quotient of Two Functions) If lim f(x) = L and lim g(x) = M, then x a x a lim [f(x)/g(x)] = L/M if M 0. x a Chapter 2: Functions, Limits and Continuity 25

26 Theorem 10. (Limit of the nth Root of a Function) If lim x a f(x) = L and n is any positive integer, then lim x a n n f(x) = L with the restriction that if n is even, L > 0. Theorem 11. For any real number a except 0 lim x a 1 x = 1 a with the restriction that if n is even, L > 0. Chapter 2: Functions, Limits and Continuity 26

27 Theorem 12. For a > 0 and n a positive integer, or if a 0 and n is an odd positive integer, then n x = n a. Theorem 13. Theorem 14. Theorem 15. lim x a lim f(x) = L if and only if lim [f(x) L] = 0. x a x a lim f(x) = L if and only if lim f(t + a)] = L. x a t 0 lim f(x) = L 1 and lim f(x) = L 2 implies L 1 = L 2. x a x a Chapter 2: Functions, Limits and Continuity 27

28 Exercises: Find the indicated limit. 1. lim (5x + 2) x 4 2. lim x 3 (2x 2 4x + 5) 3. lim y 1 (y3 2y 2 + 3y 4) 4. lim x 2 3x + 4 8x 1 5. lim x 1 6. lim x 2 2x + 1 x 2 3x + 4 x2 + 3x + 4 x lim x 3 8. lim z 5 9. lim x 1/3 10. lim x x 5 x z 2 25 z + 5 3x 1 9x 2 1 x x + 1 Chapter 2: Functions, Limits and Continuity 28

29 One-Sided Limits Definition 5. (Definition of Right-Hand Limit) Let f be a function defined at every number in some open interval (a, c). The limit of f(x) as x approaches a from the right is L, written as lim x a + f(x) = L if for any ɛ > 0, however small, there exists a δ > 0 such that if 0 < x a < δ then f(x) L < ɛ. Chapter 2: Functions, Limits and Continuity 29

30 x 0 Chapter 2: Functions, Limits and Continuity 30

31 Definition 6. (Definition of Left-Hand Limit) Let f be a function defined at every number in some open interval (d, a). The limit of f(x) as x approaches a from the left is L, written as lim x a + f(x) = L if for any ɛ > 0, however small, there exists a δ > 0 such that if 0 < a x < δ then f(x) L < ɛ. Chapter 2: Functions, Limits and Continuity 31

32 x 0 δ Chapter 2: Functions, Limits and Continuity 32

33 Example 17. Let sgn(x) = Find lim sgn(x), lim x 0 + x 0 1, if x < 0 0, if x = 0. 1, if x > 0. sgn(x). x 0 sgn(x) and lim Chapter 2: Functions, Limits and Continuity 33

34 Theorem 16. lim f(x) exists and is equal to L if and only if lim f(x) x a x a + and lim f(x) both exist and both are equal to L. x a Example 18. Let f(x) be the function defined by f(x) = x x. Then x lim x 0 + x = 1 and lim x x 0 = 1. Using Theorem (16), we see that x x lim x 0 does not exist. x Chapter 2: Functions, Limits and Continuity 34

35 Example 19. Let f be defined by f(x) = x + 5, if x < 3 9 x2, if 3 x 3 3 x, if 3 < x. Chapter 2: Functions, Limits and Continuity 35

36 Exercises: Sketch the graph of the function and find the indicated limit if it exists; if it does not exist, state the reason. 1. g(t) = 3 + t 2, if t < 2 0, if t = 2 11 t 2, if t > 2 (a) (b) (c) lim g(t) t 2 + lim t 2 g(t) lim g(t) t 2 Chapter 2: Functions, Limits and Continuity 36

37 2. f(x) = 2x + 3, if x < 1 4, if x = 1 x 2 + 2, if x > 1 (a) lim f(x) x 1 + (b) lim f(x) x 1 (c) lim f(x) x 1 3. f(x) = x + 1, if x < 1 x 2, if 1 x 1 2 x, if x > 1 (a) (b) (c) lim f(x) x 1 + lim x 1 f(x) lim f(x) x 1 (d) lim f(x) x 1 + (e) lim f(x) x 1 (f) lim f(x) x 1 Chapter 2: Functions, Limits and Continuity 37

38 More Exercise Find each of the following for the given function and specified a value. 1. f(x) = 4. f(x) = lim x a 2, if x < 1 1, if x = 1 3, if 1 < x. a = 1 { x + 4, if x 4 2. f(x) = 4 x, otherwise. a = 4 { x 2, if x 2 3. f(x) = 8 2x, otherwise. a = 2 a = 1 5. f(x) = 2x + 3, if x < 1 2, if x = 1 7 2x, if 1 < x. x 2 4, if x < 2 4, if x = 2 4 x 2, if 2 < x. f(x), lim f(x), and lim f(x) + x a ; x a ; ; ; ; a = 2 6. f(x) = x 5 ; a = 0 2, if x < 2 4 x2, if 2 x 2 7. f(x) = 2, if 2 < x. a = 2 and a = 2 x2 9, if x 3 8. f(x) = 9 x2, if 3 < x < 3 x2 9, if 3 x. a = 3 and a = 3 3 x + 1, 9. f(x) = a = 1 and a = 1 if x 1 1 x2, if 1 < x < 1 ; 3 x 1, if 1 x. Chapter 2: Functions, Limits and Continuity 38 ; ;

39 Infinite Limits One-sided infinite limits 1 Example 20. Find lim x 1 + x 1 and lim x 1 1 x 1. Chapter 2: Functions, Limits and Continuity 39

40 Two-sided infinite limits 1 Example 21. Find lim x 0 x 2. Chapter 2: Functions, Limits and Continuity 40

41 Definition 7. Infinite Limits 1. Let f be a function defined at every number in some open interval containing the number a except possibly at the number a itself. As x approaches a, f(x) increases without bound, which is written lim x a f(x) = +, if for any number N > 0 there exists a δ > 0 such that if 0 < x a < δ then f(x) > N. 2. Let f be a function defined at every number in some open interval containing the number a except possibly at the number a itself. As x approaches a, f(x) decreases without bound, which is written lim x a f(x) =, if for any number N < 0 there exists a δ > 0 such that if 0 < x a < δ then f(x) < N. Chapter 2: Functions, Limits and Continuity 41

42 Example 22. f(x) = 2x x 1 Chapter 2: Functions, Limits and Continuity 42

43 Theorem 17. If r is any positive integer, then (i) (ii) 1 lim x 0 + x r = + 1 lim x 0 x r = {, if r is odd +, if r is even. Example 23. Find (a) (b) (c) 1 lim x 0 + x 3 1 lim x 0 x 5 1 lim x 0 x 6 Chapter 2: Functions, Limits and Continuity 43

44 Theorem 18. If a is any real number and if lim x a f(x) = 0 and lim x a g(x) = c, where c is a constant not equal to 0, then (i) if c > 0 and if f(x) 0 through positive values of f(x), then lim x a g(x) f(x) = + (ii) if c > 0 and if f(x) 0 through negative values of f(x), then lim x a g(x) f(x) = (iii) if c < 0 and if f(x) 0 through positive values of f(x), then lim x a g(x) f(x) = (iv) if c < 0 and if f(x) 0 through negative values of f(x), then lim x a g(x) f(x) = + Chapter 2: Functions, Limits and Continuity 44

45 The theorem is also valid if x a is replaced by x a + or x a. Example 24. Find lim x 1 2x x 1 and lim x 1 + 2x x 1 Example 25. Find lim x 3 + x 2 + x + 2 x 2 2x 3 and lim x 3 x 2 + x + 2 x 2 2x 3 x2 4 Example 26. Let f(x) = and g(x) = x 2 Find (a) lim f(x), (b) lim g(x) x 2 + x 2 4 x 2 x 2. Chapter 2: Functions, Limits and Continuity 45

46 Example 27. Let F (x) = x2 + x + 2 x 2 2x 3. Find 1. lim x 3 + F (x) 3. lim x 1 + F (x) 2. lim x 3 F (x) 4. lim x 1 F (x) Chapter 2: Functions, Limits and Continuity 46

47 Theorem 19. (i) If lim f(x) = + and lim g(x) = c where c is any constant, then x a x a [f(x) + g(x)] = +. lim x a (ii) If lim f(x) = and lim g(x) = c where c is any constant, then x a x a [f(x) + g(x)] =. lim x a The theorem holds if x a is replaced by x a + or x a. Chapter 2: Functions, Limits and Continuity 47

48 Theorem 20. If lim f(x) = + and lim g(x) = c where c is any x a x a constant except 0, then (i) if c > 0, lim x a f(x) g(x) = + ; (ii) if c < 0, lim x a f(x) g(x) =. The theorem holds if x a is replaced by x a + or x a. Chapter 2: Functions, Limits and Continuity 48

49 Theorem 21. If lim f(x) = and lim g(x) = c where c is any x a x a constant except 0, then (i) if c > 0, lim x a f(x) g(x) = ; (ii) if c < 0, lim x a f(x) g(x) = +. The theorem holds if x a is replaced by x a + or x a. Chapter 2: Functions, Limits and Continuity 49

50 Definition 8. The line x = a is a vertical asymptote of the graph of the function f if at least one of the following statements is true. (i) (ii) (iii) (iv) lim f(x) = + x a + lim f(x) = x a + lim f(x) = + x a lim f(x) = x a Chapter 2: Functions, Limits and Continuity 50

51 Example 28. Find the vertical asymptotes of the graph of the function and sketch the graph. (a) f(x) = 2 x 4 (b) f(x) = (c) f(x) = 5 x 2 + 8x (x 4) 2 Chapter 2: Functions, Limits and Continuity 51

52 Limits at Infinity Consider the function f(x) = 2x2 x Observe that as x increases (decreases) without bound, the value of the function approaches 2. Thus, we write lim f(x) = 2 and lim x + f(x) = 2. x Chapter 2: Functions, Limits and Continuity 52

53 Definition 9. (Definition of the Limit of f(x) as x Increases without Bound) Let f be a function defined at every number in some interval (a, + ). The limit of f(x) as x increases without bound is L, written as lim f(x) = L x + if for any ɛ > 0, however small, there exists a number N > 0 such that if x > N then f(x) L < ɛ. Chapter 2: Functions, Limits and Continuity 53

54 Definition 10. (Definition of the Limit of f(x) as x Decreases without Bound) Let f be a function defined at every number in some interval (, a). The limit of f(x) as x decreases without bound is L, written as lim f(x) = L x if for any ɛ > 0, however small, there exists a number N < 0 such that if x < N then f(x) L < ɛ. Chapter 2: Functions, Limits and Continuity 54

55 Theorem 22. If r is any positive integer, then (i) lim x + 1 x r = 0 Example 29. Find lim x + 4x 3 2x + 5. Example 30. Find lim x + 2x 2 x + 5 4x 3 1. (ii) lim x Example 31. Find lim x + 3x + 4 2x2 5 and lim x x 2 Example 32. Find lim x + x x x 2 Example 33. Find lim x + 3x x r = 0 3x + 4 2x2 5. Chapter 2: Functions, Limits and Continuity 55

56 Horizontal Asymptotes Definition 11. (Definition of a Horizontal Asymptote) The line y = b is a horizontal asymptote of the graph of the function f if at least one of the following statements is true: (i) lim x + f(x) = b and for some number N, if x > N, then f(x) b; (ii) lim x f(x) = b and for some number N, if x < N, then f(x) b. Example 34. Find the horizontal asymptotes of the graph of the function x f(x)) = x Chapter 2: Functions, Limits and Continuity 56

57 Chapter 2: Functions, Limits and Continuity 57

58 Oblique Asymptotes Definition 12. (Definition of a function Continuous at a Number) The graph of the function f has the line y = mx + b as an asymptote if either of the following statements is true: (i) lim x [f(x) (mx + b)] = 0 and for some number M > 0, f(x) mx + b whenever x > M; (ii) lim x [f(x) (mx + b)] = 0 and for some number M < 0, f(x) mx + b whenever x < M. Part (i) of the definition indicates that for any ɛ > 0, there exists a number N > 0 such that if x > N then 0 < f(x) (mx + b) < ɛ, that is, we can make the function value f(x) as close to the value f(x) as Chapter 2: Functions, Limits and Continuity 58

59 close to the value of mx + b as we please by taking x sufficiently large. A similar statement may be made for part (ii) of the definition. The graph of a rational function P (x), where the degree of the Q(x) polynomial P (x) is one more than the degree of Q(x) and Q(x) is not a factor of P (x), has an oblique asymptote. To show this, we let f(x) = P (x) Q(x) and divide P (x) by Q(x) to express f(x) as the sum of a linear function and a rational function; that is, f(x) = mx + b + R(x) Q(x) where the degree of the polynomial is less than the degree of Q(x). Then f(x) (mx + b) = R(x) Q(x). Chapter 2: Functions, Limits and Continuity 59

60 When the numerator and denominator of R(x)/Q(x) is divided by the highest power of x appearing in Q(x), there will be a constant term in the denominator and all other terms in the denominator and every term in the numerator will be of the form k/x r where k is a constant and r is a positive integer. Therefore, as x +, the limit of the numerator will be zero and R(x) the limit of the denominator will be a constant. Thus, lim x Q(x) = 0. Hence, lim [f(x) (mx + b) = 0. x From this we conclude that the line y = mx + b is an oblique asymptote of the graph of f. Chapter 2: Functions, Limits and Continuity 60

61 Example 35. Find the asymptotes of the graph of h(x) = x2 + 3 x 1. Chapter 2: Functions, Limits and Continuity 61

62 Continuity Definition 13. (Definition of a function Continuous at a Number) The function f is said to be continuous at the number a if and only if the following three conditions are satisfied: (i) f(a) exists; (ii) lim x a f(x) exists; (iii) lim x a f(x) = f(a). If one or more of these three conditions fails to hold at a, the function f is said to be discontinuous at a. Chapter 2: Functions, Limits and Continuity 62

63 Continuity of the function f(x) = x + 1 at a = 0 Chapter 2: Functions, Limits and Continuity 63

64 Example 36. Consider the function f(x) = x + 1 where x 0. Then f is discontinuous at a = 0 since f(0) does not exist. Chapter 2: Functions, Limits and Continuity 64

65 { x + 1, if x 0 Example 37. Consider the function f(x) = 2, otherwise. x 0. Then f is discontinuous at a = 0 since f(0) lim x a f(x). where Removable Discontinuity Chapter 2: Functions, Limits and Continuity 65

66 Remark: The discontinuity described in each of the previous examples is called a removable discontinuity for the reason that the function can be redefined so that f(0) = 1. In general, if f is a function discontinuous at the number a but for which lim x a f(x) exists. Then either f(a) does not exist or else lim x a f(x) f(a). Such discontinuity is a removable discontinuity because f is redefined at a so that f(a) is equal to lim x a f(x), the new function becomes continuous at a. If the discontinuity is not removable, it is called an essential discontinuity. Chapter 2: Functions, Limits and Continuity 66

67 Example 38. The function f(x) = 1 is discontinuous at a = 0. x2 Infinite Discontinuity The discontinuity of this function at the number a = 0 is essential since lim x a f(x) does not exist. This kind of discontinuity is called an infinite discontinuity. Chapter 2: Functions, Limits and Continuity 67

68 Example 39. The function f(x) = a = 0. { 1, if x 0 1, if x < 0 is discontinuous at Jump Discontinuity The discontinuity illustrated in this example is essential since lim x a + f(x) lim x a f(x) and hence, lim x a f(x) does not exist. This discontinuity is called a jump discontinuity. Chapter 2: Functions, Limits and Continuity 68

69 Theorem 23. If f and g are two functions continuous at the number a, then (i) f + g is continuous at a; (ii) f g is continuous at a; (iii) f g is continuous at a; (iv) f/g is continuous at a provided that g(a) 0. Theorem 24. A polynomial function is continuous at every number. Theorem 25. A rational function is continuous at every number in its domain. Chapter 2: Functions, Limits and Continuity 69

70 Theorem 26. If n is a positive integer and then f(x) = n x 1. if n is odd, f is continuous at every number. 2. if n is even, f is continuous at every positive number. Theorem 27. (Alternative Definition of Continuity) If the function f is continuous at the number a if f is defined on some open interval containing a and if for any ɛ > 0 there exists a δ > 0 such that if x a < δ then f(x) f(a) < ɛ. Chapter 2: Functions, Limits and Continuity 70

71 Theorem 28. (Limit of a Composite Function) If lim x a g(x) = b and if the function f is continuous at b, or equivalently,. lim (f g)(x) = f(b) x a lim f(g(x)) = f( lim g(x)) x a x a Theorem 29. (Continuity of a Composite Function) If the function g is continuous at a and the function f is continuous at g(a), then the composite function f g is continuous at a. Chapter 2: Functions, Limits and Continuity 71

72 Definition 14. (Definition of Continuity on an Open Interval) A function is said to be continuous on an open interval if and only if it is continuous at every number in the open interval. Definition 15. (Definition of Right-Hand Continuity) A function is said to be continuous from the right at the number a if and only if the following three conditions are satisfied: 1. f(a) exists; 2. lim f(x) exists; x a + 3. lim f(x) = f(a). x a + Chapter 2: Functions, Limits and Continuity 72

73 Definition 16. (Definition of Left-Hand Continuity) A function is said to be continuous from the left at the number a if and only if the following three conditions are satisfied: 1. f(a) exists; 2. lim f(x) exists; x a 3. lim f(x) = f(a). x a Chapter 2: Functions, Limits and Continuity 73

74 Chapter 2: Functions, Limits and Continuity 74

75 Definition 17. (Definition of Continuity on a Closed Interval) A function whose domain includes the closed interval [a, b] is said to be continuous on [a, b] if and only if it is continuous on the open interval (a, b), as well as continuous from the right at a and continuous from the left at b. In general, a function f is right-continuous (continuous from the right) at a point x = c in its domain if lim x c + f(x) = f(c). It is leftcontinuous (continuous from the left) at a point x = c in its domain if lim x c f(x) = f(c). Thus, a function is continuous at a left endpoint a of its domain if it is right-continuous at a and continuous at a right endpoint b of its domain if it is left-continuous at b. A function is continuous at an interior point c of its domain if and only if it is both right-continuous and left-continuous at c. Chapter 2: Functions, Limits and Continuity 75

76 Example 40. The function f(x) = 4 x 2 is continuous at every point of its domain [ 2, 2]. This includes x = 2, where f is right-continuous and x = 2, where f is left-continuous. { 1, if x 0 Example 41. The unit function U(x) =, is rightcontinuous at x = 0, but is neither left-continuous nor continuous 0, if x < 0 there. Chapter 2: Functions, Limits and Continuity 76

77 Continuity Test A function f(x) is continuous at x = a if and only if it meets the following three conditions: 1. f(a) exists (c lies in the domain of f) 2. lim x a f(x) exists (f has a limit as x a) 3. lim x a f(x) = f(a) (the limit equals the function value) For one-sided continuity and continuity at an endpoint, the limits in (2) and (3) of the test should be replaced by the appropriate one-sided limits. Chapter 2: Functions, Limits and Continuity 77

78 Example 42. Consider the function y = f(x) in the given figure, whose domain is the closed interval [0, 4]. Discuss the continuity of f at x = 0, 1, 2, 3, 4. Chapter 2: Functions, Limits and Continuity 78

79 y = 4 x 2 Continuous on [ 2, 2] Chapter 2: Functions, Limits and Continuity 79

80 y = 1 x Continuous on (, 0) and (0, + ) Chapter 2: Functions, Limits and Continuity 80

81 y = U(x) Continuous on (, 0) and [0, + ) Chapter 2: Functions, Limits and Continuity 81

82 y = cos x Continuous on (, + ) Chapter 2: Functions, Limits and Continuity 82

83 Definition 18. (Definition of Continuity on a Half-Open Interval) (i) A function whose domain includes the interval half-open interval to the right [a, b) is continuous on [a, b) if it is continuous on the open interval (a, b) and continuous from the right at a. (ii) A function whose domain includes the interval half-open interval to the left (a, b] is continuous on (a, b] if it is continuous on the open interval (a, b) and continuous from the left at b. Chapter 2: Functions, Limits and Continuity 83

84 Example 43. Determine the largest interval (or union of intervals) on which the following function is continuous: f(x) = 25 x 2 x 3 Chapter 2: Functions, Limits and Continuity 84

85 The Intermediate Value Theorem Theorem 30. (The Intermediate Value Theorem (IVT).) If the function f is continuous on the closed interval [a, b], and if f(a) f(b), then for any number k between f(a) and f(b) there exists a number c between a and b such that f(c) = k. Chapter 2: Functions, Limits and Continuity 85

86 In terms of geometry, the IVT states that the graph of a continuous function on a closed interval must intersect every horizontal line y = k between the lines y = f(a) and y = f(b) at least once. The IVT also assures us that if the function is continuous on the closed interval [a, b], then f(x) assumes every value between f(a) and f(b) as x assumes values between a and b. Chapter 2: Functions, Limits and Continuity 86

87 The following is a direct consequence of the IVT. Theorem 31. (The Intermediate-Zero Theorem.) If the function f is continuous on the closed interval [a, b], and if f(a) and f(b) have opposite signs, then there exists a number c between a and b such that f(c) = 0; that is, c is a zero of f. Chapter 2: Functions, Limits and Continuity 87

88 Continuity of the Trigonometric Functions and the Squeeze Theorem Theorem 32. (The Squeeze Theorem.) Suppose that the functions f, g, and h are defined on some open interval I containing a except possibly at a itself, and that f(x) g(x) h(x) for all x I for which x a. Also suppose that lim f(x) and lim h(x) x a x a exist and are equal to L. Then lim g(x) an is equal to L. x a Chapter 2: Functions, Limits and Continuity 88

89 Example 44. Let the functions f, g and h be defined by f(x) = 4(x 2) 2 +3, g(x) = (x 2)(x2 4x + 7), h(x) = 4(x 2) (x 2) Chapter 2: Functions, Limits and Continuity 89

90 Example 45. Given g(x) 2 3(x 1) 2 for all x. Theorem to find lim g(x). x 2 Example 46. Use the Squeeze Theorem to prove that lim x 0 Use the Squeeze x sin 1 x = 0. Chapter 2: Functions, Limits and Continuity 90

91 Consider the function f(x) = sin x. This function is not defined at x x = 0 but lim f(x) exists and is equal to 1. x 0 Chapter 2: Functions, Limits and Continuity 91

92 Theorem 33. lim t 0 sin t t Proof. First assume that 0 < t < π/2. The figure shows the unit circle x 2 + y 2 = 1 and the shaded sector BOP, where B is the point (1, 0) and P is the point (cos t, sin t). The area of a circular sector of radius r and central angle of radian measure t is determined by 1 2 r2 t; so if S square units is the area of sector BOP, then S = 1 2 t. Consider now the triangle BOP with area K 1 square units. Hence, K 1 = 1 2 AP OB = 1 2 sin t. = 1 Chapter 2: Functions, Limits and Continuity 92

93 If K 2 is the area of right triangle BOT, where T is the point (1, tan t), then K 2 = 1 2 BT OB = 1 2 tan t. Observe that K 1 < S < K 2 from which we obtain the inequality 1 2 sin t < 1 2 t < 1 2 tan t. Multiplying each member of this inequality by 2/ sin t, which is positive because 0 < t < π/2, we obtain 1 < t sin t < 1 cos t. By taking the reciprocal of each member of this inequality, obtain 1 > sin t t > cos t. Chapter 2: Functions, Limits and Continuity 93

94 From cos t < sin t < 1 we obtain the inequality sin t < t. By replacing t t by 1 2t, we obtain ( ) 1 sin 2 t < 1 2 t. Squaring both sides of this inequality will yield sin 2 ( 1 2 t ) < 1 4 t2 or Thus, we obtain Using this inequality and 1 cos t 2 < 1 4 t t2 < cos t. cos t < sin t t Chapter 2: Functions, Limits and Continuity 94

95 implies t2 < sin t < 1 if 0 < t < π/2. t Now if π/2 < t < 0, then 0 < t < π/2 so that from the above inequality, we obtain ( t)2 < sin( t) ( t) < 1 if π/2 < t < 0. Hence, since sin( t) = sin t, we have t2 < sin t t < 1 if π/2 < t < 0. Therefore, t2 < sin t t < 1 if π/2 < t < π/2 and t 0. Chapter 2: Functions, Limits and Continuity 95

96 Since lim(1 1 t 2 t2 ) = 1 and lim 1 = 1 t and by applying the Squeeze Theorem we obtain the desired result lim t 0 sin t t = 1. Example 47. Find lim x 0 sin 3x sin 5x. Chapter 2: Functions, Limits and Continuity 96

97 Theorem 34. The sine function is continuous at 0. Theorem 35. The cosine function is continuous at 0. Theorem 36. lim t 0 1 cos t t = 0 Example 48. Find lim x 0 1 cos x sin x. Example 49. Find lim x 0 2 tan x x 2. Theorem 37. The sine and cosine functions are continuous at every real number. Theorem 38. The tangent, cotangent, secant, and cosecant functions are continuous on their domains. Chapter 2: Functions, Limits and Continuity 97