mathematics, calculus, limits

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.)