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- 1. Limits I
- 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)). (x, f(x)) x y = x2–2x+2
- 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)). (x+h, f(x+h) (x, f(x)) x x + h f(x+h)–f(x) h y = x2–2x+2
- 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)). (x+h, f(x+h) (x, f(x)) x x + h f(x+h)–f(x) h y = x2–2x+2
- 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. (x+h, f(x+h) (x, f(x)) x x + h f(x+h)–f(x) h y = x2–2x+2 slope = 2x–2+h
- 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, (x+h, f(x+h) (x, f(x)) x x + h f(x+h)–f(x) h y = x2–2x+2 slope = 2x–2+h
- 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 (x+h, f(x+h) (x, f(x)) x x + h f(x+h)–f(x) h y = x2–2x+2 slope = 2x–2+h
- 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. (x+h, f(x+h) (x, f(x)) x x + h f(x+h)–f(x) h y = x2–2x+2 slope = 2x–2+h
- 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 “shrinks” to 0. (x+h, f(x+h) (x, f(x)) x x + h f(x+h)–f(x) h y = x2–2x+2 slope = 2x–2+h clarify this procedure of obtaining slopes . We use the language of “limits” to
- 10. Let’s clarify the notion of “x approaches 0 from the + (right) side”. Limits I
- 11. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. Limits I
- 12. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. Limits I 0 x’s
- 13. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. Limits I 0 x’s
- 14. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. Limits I 0 x’s
- 15. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. Limits I 0 x’s ϵ for any ϵ > 0
- 16. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. Limits I 0 x’s ϵ only finitely x’s are outside for any ϵ > 0
- 17. The point here is that no matter how small the interval (0, ϵ) is, most of the x’s are in (0, ϵ). We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. Limits I 0 x’s ϵ only finitely x’s are outside for any ϵ > 0
- 18. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. 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”. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. 0 x’s ϵ only finitely x’s are outside for any ϵ > 0 Limits I
- 19. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the right” or “xi 0+” where i = 1, 2, 3… Let’s clarify the notion of “x approaches 0 from the + (right) side”. 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| / x 1” means that for any sequence xi 0+ we get |x| / x 1. We write this as lim |x| / x = 1 or lim |x| / x = 1. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (0, ϵ)”. 0+ 0 x’s ϵ only finitely x’s are outside for any ϵ > 0 Limits I x 0+
- 20. Similarly we define “x approaches 0 from the – (left) side”. Limits I
- 21. Similarly we define “x approaches 0 from the – (left) side”. Limits I We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
- 22. Similarly we define “x approaches 0 from the – (left) side”. “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 Limits I 0 x’s –ϵ We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3…
- 23. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3… Similarly we define “x approaches 0 from the – (left) side”. “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 Limits I 0 x’s –ϵ 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. We say the sequence {xi} = {x1, x2, x3, ..} “goes to 0 from the left” or “xi 0–” where i = 1, 2, 3… Similarly we define “x approaches 0 from the – (left) side”. “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 Limits I 0 x’s –ϵ 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| / x –1” means that for any sequence xi 0– we’ve |x| / x –1. We write this as lim |x| / x = –1 or lim |x| / x = –1. 0– x 0–
- 25. 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)”. Limits I
- 26. 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)”. 0 x’s –ϵ only finitely many x’s are outside x’s ϵ Limits I
- 27. Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… 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”. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”. 0 x’s –ϵ only finitely many x’s are outside x’s ϵ Limits I
- 28. Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… 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| / x is undefined because its signs are erratic if the signs of the x’s are erratic. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”. 0 x’s –ϵ only finitely many x’s are outside x’s ϵ 0 Limits I
- 29. Finally we say that “xi goes to 0” or “xi 0” where i = 1, 2, 3… 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| / x is undefined because its signs are erratic if the signs of the x’s are erratic. The direction of the x’s approaching 0 is important. “if for any number ϵ > 0 only finitely many of the x’s are outside of the neighborhood (–ϵ, ϵ)”. 0 x’s –ϵ only finitely many x’s are outside x’s ϵ 0 Limits I
- 30. Keep in mind the following examples: x’s Limits I lim |x| / x = 1 0 x–> 0+ lim |x| / x = –1 x–> 0– x’s 0 lim |x| / x = UDF x–> 0+ 0 x’s x’s
- 31. 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 )." Limits I
- 32. 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 x’s a+ϵ Limits I
- 33. 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 x’s a+ϵ 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 – ϵ, a )."
- 34. 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 x’s a+ϵ a x’s a–ϵ 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 – ϵ, a )."
- 35. 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 x’s a+ϵ a x’s a–ϵ 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 – 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. 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 x’s a+ϵ a x’s a–ϵ 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 – 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 ).” a x’s a–ϵ a+ϵ x’s
- 37. 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 x’s a+ϵ a x’s a–ϵ 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 – 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 ).” a x’s a–ϵ a+ϵ x’s We say lim f(x) = L if f(xi) L for every xi a (or a±). a (or a±)
- 38. Limits I Rules on limits: Given that all the limits exist as x a, a. lim f(x) ± g(x) = lim f(x) ± lim g(x) b. lim f(x)*g(x) = lim f(x)*lim g(x) c. lim f(x)/g(x) = lim f(x)/lim g(x) (lim g(x) ≠ 0)
- 39. The following limits are obvious. Limits I Rules on limits: Given that all the limits exist as x a, a. lim f(x) ± g(x) = lim f(x) ± lim g(x) b. lim f(x)*g(x) = lim f(x)*lim g(x) c. lim f(x)/g(x) = lim f(x)/lim g(x) (lim g(x) ≠ 0)
- 40. The following limits are obvious. * lim c = c where c is any constant. x→a (e.g lim 5 = 5) Limits I x→ a Rules on limits: Given that all the limits exist as x a, a. lim f(x) ± g(x) = lim f(x) ± lim g(x) b. lim f(x)*g(x) = lim f(x)*lim g(x) c. lim f(x)/g(x) = lim f(x)/lim g(x) (lim g(x) ≠ 0)
- 41. The following limits are obvious. * lim c = c where c is any constant. x→a * lim x = a (e.g lim 5 = 5) (e.g. lim x = 5) Limits I x→ a x→ a x→ 5 Rules on limits: Given that all the limits exist as x a, a. lim f(x) ± g(x) = lim f(x) ± lim g(x) b. lim f(x)*g(x) = lim f(x)*lim g(x) c. lim f(x)/g(x) = lim f(x)/lim g(x) (lim g(x) ≠ 0)
- 42. The following limits are obvious. * lim c = c where c is any constant. x→a * lim x = a * lim cx = ca where c is any number. (e.g lim 5 = 5) (e.g. lim x = 5) (e.g. lim 3x = 15) Limits I x→ a x→ a x→ a x→ 5 x→ 5 Rules on limits: Given that all the limits exist as x a, a. lim f(x) ± g(x) = lim f(x) ± lim g(x) b. lim f(x)*g(x) = lim f(x)*lim g(x) c. lim f(x)/g(x) = lim f(x)/lim g(x) (lim g(x) ≠ 0)
- 43. * lim (xp) = (lim x)p = ap provided ap is well defined. The following limits are obvious. * lim c = c where c is any constant. x→a * lim x = a * lim cx = ca where c is any number. (e.g lim 5 = 5) (e.g. lim x = 5) (e.g. lim x½ = 5) (e.g. lim 3x = 15) Limits I x→ a x→ a x→ a x→ a x→ 5 x→ 5 x→ a x→ 25 Rules on limits: Given that all the limits exist as x a, a. lim f(x) ± g(x) = lim f(x) ± lim g(x) b. lim f(x)*g(x) = lim f(x)*lim g(x) c. lim f(x)/g(x) = lim f(x)/lim g(x) (lim g(x) ≠ 0)
- 44. * lim (xp) = (lim x)p = ap provided ap is well defined. The following limits are obvious. * lim c = c where c is any constant. x→a * lim x = a * lim cx = ca where c is any number. (e.g lim 5 = 5) (e.g. lim x = 5) (e.g. lim x½ = 5) (e.g. lim 3x = 15) * The same statements hold true for x a±. Limits I x→ a x→ a x→ a x→ a x→ 5 x→ 5 x→ a x→ 25 Rules on limits: Given that all the limits exist as x a, a. lim f(x) ± g(x) = lim f(x) ± lim g(x) b. lim f(x)*g(x) = lim f(x)*lim g(x) c. lim f(x)/g(x) = lim f(x)/lim g(x) (lim g(x) ≠ 0)
- 45. Let P(x) and Q(x) be polynomials. Limits of Polynomial and Rational Formulas I Limits I
- 46. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I Limits I
- 47. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 2. lim = P(x) Q(x) P(a) Q(a) , Q(a) = 0. Limits I a
- 48. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 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
- 49. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 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), Limits I a
- 50. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 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±, Limits I a
- 51. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 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. Limits I a
- 52. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 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. – Limits I a
- 53. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 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 Limits I a
- 54. Let P(x) and Q(x) be polynomials. 1. lim P(x) = P(a) a Limits of Polynomial and Rational Formulas I 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
- 55. Approaching ∞ Limits I
- 56. Let’s use the function f(x) = 1/x as an example for defining the phrase “approaching ∞”. Approaching ∞ Limits I
- 57. 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. Approaching ∞ Limits I
- 58. 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. Approaching ∞ Limits I
- 59. 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 ? Approaching ∞ Limits I
- 60. 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 ∞. Approaching ∞ Limits I
- 61. 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. Approaching ∞ Limits I
- 62. 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. Limits I
- 63. 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. R x’s Limits I
- 64. 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. A set of numbers S = {x’s} is said to be bounded below if there is a number L such that L < x for all the x in the set. R x’s Limits I
- 65. 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. A set of numbers S = {x’s} 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. R x’s L x’s Limits I
- 66. 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. A set of numbers S = {x’s} 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. R x’s L x’s A set of numbers S = {x’s} is bounded if it’s bounded above and below. R x’s L Limits I
- 67. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. Limits I
- 68. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. Limits I
- 69. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. This list has the following property. Limits I
- 70. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. This list has the following property. For any large number G we select, there are only finitely many entries that are smaller than G. Limits I
- 71. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. 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. Limits I
- 72. x 0.1 0.01 0.001 0.0001 0+ f(x) = 1/x 10 100 1,000 10,000 … The 1/x–values on the list is bounded below – a lower bound L = 0 < 1/x. However the list is not bounded above. 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 = 10100 < all entries only these entries are < 10100 Limits I
- 73. In the language of limits, we say that x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 10100 < all entries lim 1/x = ∞ 0+ Limits I
- 74. In the language of limits, we say that x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 10100 < all entries lim 1/x = ∞ and it is read as “the limit of 1/x, as x goes to 0+ is ∞”. 0+ Limits I
- 75. In the language of limits, we say that x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 10100 < all entries lim 1/x = ∞ 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– 0+ Limits I
- 76. In the language of limits, we say that x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 10100 < all entries lim 1/x = ∞ 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– 0+ However lim 1/x is undefined (UDF) because the signs of 1/x is unknown so no general conclusion may be made except that |1/x| ∞. 0 Limits I
- 77. In the language of limits, we say that x 0.1 0.01 0.001 0.0001 … 100th entry … f(x) = 1/x 10 100 1,000 10,000 … G = 10100 < all entries lim 1/x = ∞ 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– 0+ However lim 1/x is undefined (UDF) because the 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. 0 Limits I
- 78. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, Limits I
- 79. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also Limits I say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L.
- 80. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞
- 81. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ “boundary behaviors” of 1/x.
- 82. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ “boundary behaviors” of 1/x. The domain of 1/x is x = 0 so its “boundary” consists of the following.
- 83. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ “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.
- 84. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ 0+ As x 0+, lim 1/x = ∞ “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.
- 85. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x 0+, lim 1/x = ∞ “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.
- 86. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x 0+, lim 1/x = ∞ “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. y = 1/x x= 0: Vertical Asymptote
- 87. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x 0+, lim 1/x = ∞ “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. y = 1/x x= 0: Vertical Asymptote
- 88. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x 0+, lim 1/x = ∞ “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. y = 1/x x= 0: Vertical Asymptote
- 89. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x 0+, lim 1/x = ∞ “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. y = 1/x x= 0: Vertical Asymptote
- 90. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x 0+, lim 1/x = ∞ “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. y = 1/x x= 0: Vertical Asymptote
- 91. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ As x 0+, lim 1/x = ∞ “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. II. The two “ends” of the line. y = 1/x x= 0: Vertical Asymptote
- 92. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ As x 0+, lim 1/x = ∞ “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. II. The two “ends” of the line. y = 1/x x= 0: Vertical Asymptote
- 93. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ As x 0+, lim 1/x = ∞ “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. II. The two “ends” of the line. y = 1/x x= 0: Vertical Asymptote
- 94. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ As x 0+, lim 1/x = ∞ “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. II. The two “ends” of the line. y = 1/x x= 0: Vertical Asymptote
- 95. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ As x 0+, lim 1/x = ∞ “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. II. The two “ends” of the line. y = 1/x x= 0: Vertical Asymptote
- 96. Similarly we say that “as x ∞, f(x) L” if for every sequence xi ∞ we have that f(xi) L, and we also lim 1/x = lim 1/x = 0. Let’s summarize the ∞ Limits I Hence say that “as x –∞, f(x) L” if for every sequence xi –∞ we have that f(xi) L. –∞ As x 0–, lim 1/x = –∞ 0+ 0– As x ∞, lim 1/x = 0+ ∞ As x –∞, lim 1/x = 0– –∞ As x 0+, lim 1/x = ∞ “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. II. The two “ends” of the line. y = 1/x x= 0: Vertical Asymptote y = 0: Horizontal Asymptote
- 97. Arithmetic of ∞ Limits I
- 98. Arithmetic of ∞ Limits I The symbol “∞” is not a number because it does not represent a numerical measurement.
- 99. 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.
- 100. 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 –∞”.
- 101. 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 ∞.
- 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. 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.
- 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 –∞”. 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.
- 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 ∞. 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.
- 105. Arithmetic of ∞ Limits I
- 106. Arithmetic of ∞ Limits I 1. ∞ + ∞ = ∞ 2. ∞ * ∞ = ∞ 3. c * ∞ = ∞ for any constant c > 0. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c.
- 107. 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. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c.
- 108. 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. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c.
- 109. 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. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c. As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞.
- 110. 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. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c. As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. (Not true for “/“.)
- 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. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c. As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. As x goes to ∞, lim x = ∞, so lim 3x = ∞. (Not true for “/“.)
- 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. ∞ * ∞ = ∞ 3. c * ∞ = ∞ for any constant c > 0. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c. As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. As x goes to ∞, lim x = ∞, so lim 3x = ∞. As x goes to ∞, lim x = ∞, so lim 3/x = 0. (Not true for “/“.)
- 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. ∞ * ∞ = ∞ 3. c * ∞ = ∞ for any constant c > 0. 5. Given a fixed b, b∞ = ∞ if b > 1, b∞ = 0 if 0 < b < 1 4. c / ∞ = 0 for any constant c. As x goes to ∞, lim x = ∞ and lim x2 = ∞, so lim (x * x2) = lim x * lim x2 = ∞. As x goes to ∞, lim x = ∞, so lim 3x = ∞. As x goes to ∞, lim x = ∞, so lim 3/x = 0. As x goes to ∞, lim 2x = ∞ and lim (½)x = 0. (Not true for “/“.)
- 114. Limits I The following situations of limits are inconclusive.
- 115. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive.
- 116. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. As x goes to ∞, lim x = ∞ and lim x2 = ∞,
- 117. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x – x = lim 0 = 0, and lim x2 – x = ∞.
- 118. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 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 ∞.
- 119. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 2. ∞ / ∞ = ? (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 ∞.
- 120. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, 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 ∞.
- 121. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x / x2 = 0, 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. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 2. ∞ / ∞ = ? (inconclusive form) As x goes to ∞, lim x = ∞ and lim x2 = ∞, but lim x / x2 = 0, lim x/x = 1, 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 ∞.
- 123. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 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 = ∞. 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 ∞.
- 124. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 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 = ∞. 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 ∞. Again, all these questions are in the form ∞/∞ but have different behaviors as x ∞.
- 125. Limits I 1. ∞ – ∞ = ? (inconclusive form) The following situations of limits are inconclusive. 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 = ∞. 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 ∞. We have to find other ways to determine the limiting behaviors when a problem is in the inconclusive ∞ – ∞ and ∞ / ∞ form. Again, all these questions are in the form ∞/∞ but have different behaviors as x ∞.
- 126. Limits I For example the is of the ∞ / ∞ form as x ∞, therefore we will have to transform the formula to determine its behavior. 3x + 4 5x + 6
- 127. Limits I For example the is of the ∞ / ∞ form as x ∞, therefore we will have to transform the formula to determine its behavior. 3x + 4 5x + 6 3x + 4 5x + 6 lim = 3/5. ∞ We will talk about various methods in the next section in determining the limits of formulas with inconclusive forms and see that (Take out the calculator and try to find it.)