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### 2.1 limits i

1. 1. Limits I
2. 2. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). y = x2–2x+2 (x, f(x)) x
3. 3. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). y = x2–2x+2 (x+h, f(x+h) (x, f(x)) f(x+h)–f(x) h x x + h
4. 4. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). y = x2–2x+2 (x+h, f(x+h) (x, f(x)) f(x+h)–f(x) h x x + h
5. 5. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). We obtained the cord–slope–formula 2x – 2 + h. y = x2–2x+2 (x+h, f(x+h) slope = 2x–2+h (x, f(x)) f(x+h)–f(x) h x x + h
6. 6. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). We obtained the cord–slope–formula 2x – 2 + h. We reason that as the values of h shrinks to 0, y = x2–2x+2 (x+h, f(x+h) slope = 2x–2+h (x, f(x)) f(x+h)–f(x) h x x + h
7. 7. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). We obtained the cord–slope–formula 2x – 2 + h. We reason that as the values of h shrinks to 0, the cords slide towards the tangent line y = x2–2x+2 (x+h, f(x+h) slope = 2x–2+h (x, f(x)) f(x+h)–f(x) h x x + h
8. 8. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). We obtained the cord–slope–formula 2x – 2 + h. We reason that as the values of h shrinks to 0, the cords slide towards the tangent line so the slope at (x, f(x)) must be 2x – 2 because h “fades” to 0. y = x2–2x+2 (x+h, f(x+h) slope = 2x–2+h (x, f(x)) f(x+h)–f(x) h x x + h
9. 9. Limits I In the calculation of the derivative of f(x) = x2 – 2x + 2, we simplified the slope (difference-quotient) formula of the cords with one end fixed at (x, f(x)) and the other end at (x+h, f(x+h)). We obtained the cord–slope–formula 2x – 2 + h. We reason that as the values of h shrinks to 0, the cords slide towards the tangent line so the slope at (x, f(x)) must be 2x – 2 because h “fades” to 0. y = x2–2x+2 (x+h, f(x+h) slope = 2x–2+h (x, f(x)) f(x+h)–f(x) h x x + h We use the language of “limits” to clarify this procedure of obtaining slopes .
10. 10. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”.
11. 11. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3…
12. 12. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”.
13. 13. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. 0 x’s
14. 14. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. for any ϵ > 0 0 ϵ x’s
15. 15. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. for any ϵ > 0 only finitely x’s are outside 0 ϵ x’s
16. 16. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. for any ϵ > 0 only finitely x’s are outside 0 ϵ x’s The point here is that no matter how small the interval (0, ϵ) is, most of the x’s are in (0, ϵ).
17. 17. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. for any ϵ > 0 only finitely x’s are outside 0 ϵ x’s We say “as x goes to 0+ we get that …” we mean that for “every sequence {xi} where xi 0+ we would obtain the result mentioned”.
18. 18. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. for any ϵ > 0 only finitely x’s are outside 0 ϵ x’s We say “as x goes to 0+ we get that …” we mean that for “every sequence {xi} where xi 0+ we would obtain the result mentioned”. So “as x 0+, x + 2 2” means that for any sequence xi 0+ we get xi + 2 2.
19. 19. Limits I Let’s clarify the notion of “x approaches 0 from the + (right) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. for any ϵ > 0 only finitely x’s are outside We say “as x goes to 0+ we get that …” we mean that for “every sequence {xi} where xi 0+ we would obtain the result mentioned”. So “as x 0+, x + 2 2” means that for any sequence xi 0+ we get xi + 2 2. We write this as lim (x + 2) = 2 or lim (x + 2) = 2. 0+ 0 ϵ x’s x 0+
20. 20. Limits I Similarly we define “x approaches 0 from the – (left) side”.
21. 21. Limits I Similarly we define “x approaches 0 from the – (left) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
22. 22. Limits I Similarly we define “x approaches 0 from the – (left) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ,0)”. only finitely x’s are outside for any ϵ > 0 x’s –ϵ 0
23. 23. Limits I Similarly we define “x approaches 0 from the – (left) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ,0)”. only finitely x’s are outside for any ϵ > 0 x’s –ϵ 0 We say “as x goes to 0– we get that …” we mean that for “every sequence {xi} where xi 0– we would obtain the result mentioned”.
24. 24. Limits I Similarly we define “x approaches 0 from the – (left) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ,0)”. only finitely x’s are outside for any ϵ > 0 x’s –ϵ 0 We say “as x goes to 0– we get that …” we mean that for “every sequence {xi} where xi 0– we would obtain the result mentioned”. So “as x 0–, x + 2 2” means that for any sequence xi 0– we get xi + 2 2.
25. 25. Similarly we define “x approaches 0 from the – (left) side”. We say the sequence {xi} = {x1, x2, x3, .. } “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ,0)”. only finitely x’s are outside for any ϵ > 0 We say “as x goes to 0– we get that …” we mean that for “every sequence {xi} where xi 0– we would obtain the result mentioned”. So “as x 0–, x + 2 2” means that for any sequence xi 0– we get xi + 2 2. We write this as lim (x + 2) = 2 or lim (x + 2) = 2. 0– Limits I x 0– x’s –ϵ 0
26. 26. Limits I Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”.
27. 27. Limits I Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”. x’s –ϵ 0 ϵ x’s only finitely many x’s are outside
28. 28. Limits I Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”. x’s –ϵ 0 ϵ x’s only finitely many x’s are outside We say “as x goes to 0 we get that …” we mean that for “every sequence {xi} where xi 0 we obtain the result mentioned”.
29. 29. Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”. x’s –ϵ 0 ϵ x’s only finitely many x’s are outside We say “as x goes to 0 we get that …” we mean that for “every sequence {xi} where xi 0 we obtain the result mentioned”. Hence lim x + 1 = 1/–1 = –1. 0 2x – 1 Limits I
30. 30. Limits I Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”. x’s –ϵ 0 ϵ x’s only finitely many x’s are outside We say “as x goes to 0 we get that …” we mean that for “every sequence {xi} where xi 0 we obtain the result mentioned”. Hence lim x + 1 = 1/–1 = –1. 0 2x – 1 Let’s generalize this to “x a” where a is any number.
31. 31. Limits I The notation “xi a+” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a, a + ϵ )."
32. 32. Limits I The notation “xi a+” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a, a + ϵ )." a a+ϵ x’s
33. 33. Limits I The notation “xi a+” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a, a + ϵ )." a a+ϵ x’s The notation “xi a–” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a )."
34. 34. Limits I The notation “xi a+” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a, a + ϵ )." a a+ϵ x’s The notation “xi a–” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a )." x’s a–ϵ a
35. 35. Limits I The notation “xi a+” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a, a + ϵ )." a a+ϵ x’s The notation “xi a–” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a )." x’s a–ϵ a The notation “xi a” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a + ϵ ).”
36. 36. Limits I The notation “xi a+” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a, a + ϵ )." a a+ϵ x’s The notation “xi a–” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a )." x’s a–ϵ a The notation “xi a” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a + ϵ ).” x’s a–ϵ a a+ϵ x’s
37. 37. Limits I The notation “xi a+” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a, a + ϵ )." a a+ϵ x’s The notation “xi a–” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a )." x’s a–ϵ a The notation “xi a” where {xi} is a sequence of numbers means that “for every ϵ > 0, only finitely many x’s are outside of the interval (a – ϵ, a + ϵ ).” x’s a–ϵ a a+ϵ x’s We say lim f(x) = L if f(xi) L for every xi a (or a±). a (or a±)
38. 38. Limits I The following statements of limits as x a are true.
39. 39. The following statements of limits as x a are true. * lim c = c where c is any constant. a Limits I
40. 40. The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) Limits I a
41. 41. The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) * lim x = a Limits I a a
42. 42. The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) * lim x = a (e.g. lim x = 5) Limits I a a 5
43. 43. Limits I The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) a * lim x = a a (e.g. lim x = 5) * lim cx = ca where c is any number. a 5
44. 44. The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) * lim x = a (e.g. lim x = 5) * lim cx = ca where c is any number. (e.g. lim 3x = 15) Limits I a a a 5 5
45. 45. Limits I The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) a * lim x = a a (e.g. lim x = 5) * lim cx = ca where c is any number. a (e.g. lim 3x = 15) * lim (xp) = (lim x)p = ap provided ap is well defined. a 5 5 a
46. 46. The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) * lim x = a (e.g. lim x = 5) * lim cx = ca where c is any number. (e.g. lim 3x = 15) * lim (xp) = (lim x)p = ap provided ap is well defined. (e.g. lim x½ = 5) Limits I a a a a 5 5 a 25
47. 47. Limits I The following statements of limits as x a are true. * lim c = c where c is any constant. a (e.g lim 5 = 5) a * lim x = a a (e.g. lim x = 5) 5 * lim cx = ca where c is any number. a (e.g. lim 3x = 15) 5 * lim (xp) = (lim x)p = ap provided ap is well defined. a a (e.g. lim x½ = 5) 25 Reminder: the same statements hold true for x a±.
48. 48. Limits I Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials.
49. 49. Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits I
50. 50. Limits I Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. a
51. 51. Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. (e.g. lim x + 2 x – 3 1 = –3/2) Limits I a
52. 52. Limits I Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. (e.g. lim x + 2 x – 3 1 = –3/2) a In fact, if f(x) is an elementary function and f(a) is well defined, i.e. a is in the domain of f(x),
53. 53. Limits I Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. (e.g. lim x + 2 x – 3 1 = –3/2) a In fact, if f(x) is an elementary function and f(a) is well defined, i.e. a is in the domain of f(x), then lim f(x) = f(a) as x a or x a±,
54. 54. Limits I Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. (e.g. lim x + 2 x – 3 1 = –3/2) a In fact, if f(x) is an elementary function and f(a) is well defined, i.e. a is in the domain of f(x), then lim f(x) = f(a) as x a or x a±, provided the selections of such x’s are possible.
55. 55. Limits I Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. (e.g. lim x + 2 x – 3 1 = –3/2) a In fact, if f(x) is an elementary function and f(a) is well defined, i.e. a is in the domain of f(x), then lim f(x) = f(a) as x a or x a±, provided the selections of such x’s are possible. For example, the domain of the function f(x) = √x is 0 <– x.
56. 56. Limits I Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. (e.g. lim x + 2 x – 3 1 = –3/2) a In fact, if f(x) is an elementary function and f(a) is well defined, i.e. a is in the domain of f(x), then lim f(x) = f(a) as x a or x a±, provided the selections of such x’s are possible. For example, the domain of the function f(x) = √x is 0 <– x. Hence lim√x = √a for 0 < a. a
57. 57. Limits of Polynomial and Rational Formulas I Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. (e.g. lim x + 2 x – 3 1 = –3/2) In fact, if f(x) is an elementary function and f(a) is well defined, i.e. a is in the domain of f(x), then lim f(x) = f(a) as x a or x a±, provided the selections of such x’s are possible. For example, the domain of the function f(x) = √x is 0 <– x. Hence lim√x = √a for 0 < a. a However at a = 0, we could only have lim √x = 0 = f(0) as shown. y = x1/2 0+ (but not 0) Limits I a
58. 58. Approaching ∞ Limits I
59. 59. Approaching ∞ Limits I Let’s use the function f(x) = 1/x as an example for defining the phrase “approaching ∞”.
60. 60. Approaching ∞ Limits I Let’s use the function f(x) = 1/x as an example for defining the phrase “approaching ∞”. The domain of the 1/x is the set of all numbers x except x = 0.
61. 61. Approaching ∞ Limits I Let’s use the function f(x) = 1/x as an example for defining the phrase “approaching ∞”. The domain of the 1/x is the set of all numbers x except x = 0. Although we can’t evaluate 1/x with x = 0, we still know the behavior of f(x) as x takes on small values that are close to 0 as demonstrated in the table below.
62. 62. Approaching ∞ Limits I Let’s use the function f(x) = 1/x as an example for defining the phrase “approaching ∞”. The domain of the 1/x is the set of all numbers x except x = 0. Although we can’t evaluate 1/x with x = 0, we still know the behavior of f(x) as x takes on small values that are close to 0 as demonstrated in the table below. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 ?
63. 63. Approaching ∞ Limits I Let’s use the function f(x) = 1/x as an example for defining the phrase “approaching ∞”. The domain of the 1/x is the set of all numbers x except x = 0. Although we can’t evaluate 1/x with x = 0, we still know the behavior of f(x) as x takes on small values that are close to 0 as demonstrated in the table below. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 ? From the table we see that the corresponding 1/x expands unboundedly to ∞.
64. 64. Approaching ∞ Limits I Let’s use the function f(x) = 1/x as an example for defining the phrase “approaching ∞”. The domain of the 1/x is the set of all numbers x except x = 0. Although we can’t evaluate 1/x with x = 0, we still know the behavior of f(x) as x takes on small values that are close to 0 as demonstrated in the table below. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 ? From the table we see that the corresponding 1/x expands unboundedly to ∞. Let’s make “expands unboundedly to ∞” more precise.
65. 65. Limits I A set of infinitely many numbers S = {x’s} is said to be bounded above if there is a number R such that x < R for all the numbers x in the set S.
66. 66. Limits I A set of infinitely many numbers S = {x’s} is said to be bounded above if there is a number R such that x < R for all the numbers x in the set S. The “R” stands for “to the right” as shown. x’s R
67. 67. Limits I A set of infinitely many numbers S = {x’s} is said to be bounded above if there is a number R such that x < R for all the numbers x in the set S. The “R” stands for “to the right” as shown. x’s R A set of numbers S = {x} is said to be bounded below if there is a number L such that L < x for all the x in the set.
68. 68. Limits I A set of infinitely many numbers S = {x’s} is said to be bounded above if there is a number R such that x < R for all the numbers x in the set S. The “R” stands for “to the right” as shown. x’s R A set of numbers S = {x} is said to be bounded below if there is a number L such that L < x for all the x in the set. The “L” stands for “to the left” as shown. L x’s
69. 69. Limits I A set of infinitely many numbers S = {x’s} is said to be bounded above if there is a number R such that x < R for all the numbers x in the set S. The “R” stands for “to the right” as shown. x’s R A set of numbers S = {x} is said to be bounded below if there is a number L such that L < x for all the x in the set. The “L” stands for “to the left” as shown. L x’s We say that the interval (L, R) is bounded above and below, or that it is bounded. L x’s R
70. 70. Limits I The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 …
71. 71. Limits I The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 …
72. 72. Limits I The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … This list has the following property.
73. 73. Limits I The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … This list has the following property. For any large number G we select, there are only finitely many entries that are smaller than G.
74. 74. Limits I The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … This list has the following property. For any large number G we select, there are only finitely many entries that are smaller than G. For example, if G = 10100 then only entries to the left of the 100th entry are less than G.
75. 75. Limits I The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … This list has the following property. For any large number G we select, there are only finitely many entries that are smaller than G. For example, if G = 10100 then only entries to the left of the 100th entry are less than G. x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries only these entries are < 10100
76. 76. Limits I x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries In the language of limits, we say that lim 1/x = ∞ 0+
77. 77. Limits I x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries In the language of limits, we say that lim 1/x = ∞ 0+ and it is read as “the limit of 1/x, as x goes to 0+ is ∞”.
78. 78. Limits I x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries In the language of limits, we say that lim 1/x = ∞ 0+ and it is read as “the limit of 1/x, as x goes to 0+ is ∞”. In a similar fashion we have that “the limit of 1/x, as x goes to 0– is –∞” as lim 1/x = –∞ 0–
79. 79. x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries In the language of limits, we say that lim 1/x = ∞ 0+ and it is read as “the limit of 1/x, as x goes to 0+ is ∞”. In a similar fashion we have that “the limit of 1/x, as x goes to 0– is –∞” as lim 1/x = –∞ 0– However lim 1/x is undefined (UDF) because the 0 signs of 1/x is unknown so no general conclusion may be made except that |1/x| ∞. Limits I
80. 80. x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 1 0 1 0 0 < all entries In the language of limits, we say that lim 1/x = ∞ 0+ and it is read as “the limit of 1/x, as x goes to 0+ is ∞”. In a similar fashion we have that “the limit of 1/x, as x goes to 0– is –∞” as lim 1/x = –∞ 0– However lim 1/x is undefined (UDF) because the 0 signs of 1/x is unknown so no general conclusion may be made except that |1/x| ∞. The behavior of 1/x may fluctuate wildly depending on the selections of the x’s. Limits I
81. 81. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L,
82. 82. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L.
83. 83. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. ∞ Hence –∞
84. 84. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x.
85. 85. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following.
86. 86. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0.
87. 87. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. 0+ As x 0+, lim 1/x = ∞
88. 88. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0–
89. 89. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– y = 1/x x= 0: Vertical Asymptote
90. 90. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– y = 1/x x= 0: Vertical Asymptote
91. 91. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– y = 1/x x= 0: Vertical Asymptote
92. 92. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– y = 1/x x= 0: Vertical Asymptote
93. 93. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– y = 1/x x= 0: Vertical Asymptote
94. 94. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– II. The two “ends” of the line. y = 1/x As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ x= 0: Vertical Asymptote
95. 95. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– II. The two “ends” of the line. y = 1/x As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ x= 0: Vertical Asymptote
96. 96. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– II. The two “ends” of the line. y = 1/x As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ x= 0: Vertical Asymptote
97. 97. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– II. The two “ends” of the line. y = 1/x As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ x= 0: Vertical Asymptote
98. 98. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– II. The two “ends” of the line. y = 1/x As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ x= 0: Vertical Asymptote
99. 99. Limits I Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Hence –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following. I. The vertical asymptote x = 0. As x 0+, lim 1/x = ∞ 0+ As x 0–, lim 1/x = –∞ 0– II. The two “ends” of the line. y = 1/x As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ x= 0: Vertical Asymptote y = 0: Horizontal Asymptote
100. 100. Arithmetic of ∞ Limits I
101. 101. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement.
102. 102. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement. It represents the behavior of a list of endless numbers, specifically eventually “almost all” of the numbers are larger than any imaginable numbers.
103. 103. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement. It represents the behavior of a list of endless numbers, specifically eventually “almost all” of the numbers are larger than any imaginable numbers. Hence we say that “the sequence 1, 2, 3, .. goes to ∞” or that “the sequence –1, –2, –3, .. goes to –∞”.
104. 104. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement. It represents the behavior of a list of endless numbers, specifically eventually “almost all” of the numbers are larger than any imaginable numbers. Hence we say that “the sequence 1, 2, 3, .. goes to ∞” or that “the sequence –1, –2, –3, .. goes to –∞”. If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,.. the resulting sequence still goes to ∞.
105. 105. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement. It represents the behavior of a list of endless numbers, specifically eventually “almost all” of the numbers are larger than any imaginable numbers. Hence we say that “the sequence 1, 2, 3, .. goes to ∞” or that “the sequence –1, –2, –3, .. goes to –∞”. If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,.. the resulting sequence still goes to ∞. In fact, given any sequence of xi such that xi ∞, then cxi ∞ for any 0 < c.
106. 106. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement. It represents the behavior of a list of endless numbers, specifically eventually “almost all” of the numbers are larger than any imaginable numbers. Hence we say that “the sequence 1, 2, 3, .. goes to ∞” or that “the sequence –1, –2, –3, .. goes to –∞”. If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,.. the resulting sequence still goes to ∞. In fact, given any sequence of xi such that xi ∞, then cxi ∞ for any 0 < c. In short, we say that c* ∞ = ∞ for any constant c > 0.
107. 107. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement. It represents the behavior of a list of endless numbers, specifically eventually “almost all” of the numbers are larger than any imaginable numbers. Hence we say that “the sequence 1, 2, 3, .. goes to ∞” or that “the sequence –1, –2, –3, .. goes to –∞”. If we multiple 1, 2, 3 … by 2 so it becomes 2, 4, 6,.. the resulting sequence still goes to ∞. In fact, given any sequence of xi such that xi ∞, then cxi ∞ for any 0 < c. In short, we say that c* ∞ = ∞ for any constant c > 0. We summarize these facts about ∞ below.
108. 108. Arithmetic of ∞ Limits I
109. 109. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ 2. ∞ * ∞ = ∞ 3. c * ∞ = ∞ for any constant c > 0. 4. c / ∞ = 0 for any constant c. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
110. 110. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence lim (x + x2) = lim x + lim x2 = ∞. 2. ∞ * ∞ = ∞ 3. c * ∞ = ∞ for any constant c > 0. 4. c / ∞ = 0 for any constant c. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
111. 111. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.) 2. ∞ * ∞ = ∞ 3. c * ∞ = ∞ for any constant c > 0. 4. c / ∞ = 0 for any constant c. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
112. 112. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.) 2. ∞ * ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. 3. c * ∞ = ∞ for any constant c > 0. 4. c / ∞ = 0 for any constant c. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
113. 113. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.) 2. ∞ * ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. (Not true for “/“.) 3. c * ∞ = ∞ for any constant c > 0. 4. c / ∞ = 0 for any constant c. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
114. 114. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.) 2. ∞ * ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. 3. c * ∞ = ∞ for any constant c > 0. (Not true for “/“.) As x goes to ∞, lim x = ∞, so lim 3x = ∞. 4. c / ∞ = 0 for any constant c. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
115. 115. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.) 2. ∞ * ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. 3. c * ∞ = ∞ for any constant c > 0. (Not true for “/“.) As x goes to ∞, lim x = ∞, so lim 3x = ∞. 4. c / ∞ = 0 for any constant c. As x goes to ∞, lim x = ∞, so lim 3/x = 0. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1
116. 116. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, hence lim (x + x2) = lim x + lim x2 = ∞. (Not true for “–“.) 2. ∞ * ∞ = ∞ As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. 3. c * ∞ = ∞ for any constant c > 0. (Not true for “/“.) As x goes to ∞, lim x = ∞, so lim 3x = ∞. 4. c / ∞ = 0 for any constant c. As x goes to ∞, lim x = ∞, so lim 3/x = 0. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 As x goes to ∞, lim 2x = ∞ and lim (½)x = 0.
117. 117. Limits I The following situations of limits are inconclusive.
118. 118. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form)
119. 119. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞,
120. 120. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
121. 121. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞.
122. 122. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞. 2. ∞ / ∞ = ? (inconclusive form)
123. 123. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞,
124. 124. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x / x2 = 0,
125. 125. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x / x2 = 0, lim x/x = 1,
126. 126. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x / x2 = 0, lim x/x = 1, and lim x2/x = ∞.
127. 127. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x / x2 = 0, lim x/x = 1, and lim x2/x = ∞. Again, all these questions are in the form ∞/∞ but have different behaviors as x ∞.
128. 128. Limits I The following situations of limits are inconclusive. 1. ∞ – ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞. Both questions are in the form of ∞ – ∞ yet they have drastically different behaviors as x ∞. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x / x2 = 0, lim x/x = 1, and lim x2/x = ∞. Again, all these questions are in the form ∞/∞ but have different behaviors as x ∞. We have to find other ways to determine the limiting behaviors when a problem is in the inconclusive ∞ – ∞ and ∞ / ∞ form.
129. 129. Limits I 3x + 4 5x + 6 For example the is of the ∞ / ∞ form as x ∞, therefore we will have to transform the formula to determine its behavior.
130. 130. Limits I 3x + 4 5x + 6 For example the is of the ∞ / ∞ form as x ∞, therefore we will have to transform the formula to determine its behavior. We will talk about various methods in the next section in determining the limits of formulas with inconclusive forms and see that 3x + 4 5x + 6 lim = 3/5. ∞ (Take out the calculator and try to find it.)