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)).
y = x2–2x+2
(x, f(x))
x

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. 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. 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. 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. 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. 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. 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. Limits I
Let’s clarify the notion of
“x approaches 0 from the + (right) side”.

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. 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. 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. 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. 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. 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. 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. 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. 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. Limits I
Similarly we define
“x approaches 0 from the – (left) side”.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Limits I
The following statements of limits as x a are true.

39. The following statements of limits as x a are true.
* lim c = c where c is any constant.
a
Limits I

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. 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. 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. 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. 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. 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. 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. 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. Limits I
Limits of Polynomial and Rational Formulas I
Let P(x) and Q(x) be polynomials.

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. 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. 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. 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. 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. 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. 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. 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. 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. Approaching ∞
Limits I

59. Approaching ∞
Limits I
Let’s use the function f(x) = 1/x as an example for
defining the phrase “approaching ∞”.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Limits I
Similarly we say that “as x ∞, f(x) L” if for every
sequence xi ∞ we have that f(xi) L,

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Arithmetic of ∞
Limits I

101. Arithmetic of ∞
Limits I
The symbol “∞” is not a number because it does not
represent a numerical measurement.

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. 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. 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. 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. 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. 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. Arithmetic of ∞
Limits I

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. 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. 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. 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. 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. 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. 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. 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. Limits I
The following situations of limits are inconclusive.

118. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)

119. Limits I
The following situations of limits are inconclusive.
1. ∞ – ∞ = ? (inconclusive form)
As x goes to ∞, lim x = ∞ and lim x2 = ∞,

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.)