Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.

1. Section 1.6
Limits involving Infinity
V63.0121.034, Calculus I
September 21, 2009
Announcements
Quiz 1 this week (§§1.1–1.3)
Written Assignment 2 due Wednesday
. . . . . .

2. Recall the definition of limit
Definition
We write
lim f(x) = L
x→a
and say
“the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) but not equal to a.
. . . . . .

3. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x→0+ x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .

4. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x→0+ x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .

5. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x→0+ x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .

6. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x→0+ x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .

7. Outline
Infinite Limits
Vertical Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
. . . . . .

8. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
. x
.
. . . . . .

9. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .

10. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .

11. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .

12. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .

13. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .

14. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .

15. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x→a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .

16. Negative Infinity
Definition
The notation
lim f(x) = −∞
x→a
means that the values of f(x) can be made arbitrarily large
negative (as large as we please) by taking x sufficiently close to a
but not equal to a.
. . . . . .

17. Negative Infinity
Definition
The notation
lim f(x) = −∞
x→a
means that the values of f(x) can be made arbitrarily large
negative (as large as we please) by taking x sufficiently close to a
but not equal to a.
We call a number large or small based on its absolute value.
So −1, 000, 000 is a large (negative) number.
. . . . . .

18. Vertical Asymptotes
Definition
The line x = a is called a vertical asymptote of the curve y = f(x)
if at least one of the following is true:
lim f(x) = ∞ lim f(x) = −∞
x→a x→a
lim f(x) = ∞ lim f(x) = −∞
x→a+ x→a+
lim f(x) = ∞ lim f(x) = −∞
x→a− x→a−
. . . . . .

19. Infinite Limits we Know
y
.
.
.
1
lim
+ x
=∞
x→0 .
. . . . . . . x
.
.
.
.
. . . . . .

20. Infinite Limits we Know
y
.
.
.
1
lim
+ x
=∞
x→0 .
1
lim = −∞
x→0− x . . . . . . . x
.
.
.
.
. . . . . .

21. Infinite Limits we Know
y
.
.
.
1
lim
+ x
=∞
x→0 .
1
lim = −∞
x→0− x . . . . . . . x
.
1
lim =∞
x→0 x2 .
.
.
. . . . . .

22. Finding limits at trouble spots
Example
Let
x2 + 2
f (x ) =
x2 − 3x + 2
Find lim f(x) and lim f(x) for each a at which f is not
x→a− x→a+
continuous.
. . . . . .

23. Finding limits at trouble spots
Example
Let
x2 + 2
f (x ) =
x2 − 3x + 2
Find lim f(x) and lim f(x) for each a at which f is not
x→a− x→a+
continuous.
Solution
The denominator factors as (x − 1)(x − 2). We can record the
signs of the factors on the number line.
. . . . . .

24. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
. . . . . .

25. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
. . . . . .

26. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
. . . . . .

27. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
. . f
.(x)
1
. 2
.
. . . . . .

28. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . . f
.(x)
1
. 2
.
. . . . . .

29. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ ±.
. ∞ . f
.(x)
1
. 2
.
. . . . . .

30. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ ±. −
. ∞ . . f
.(x)
1
. 2
.
. . . . . .

31. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ ±. −
. ∞ . . ∞. f
.(x)
1
. 2
.
. . . . . .

32. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ ±. −
. ∞ . . ∞. .
+
f
.(x)
1
. 2
.
. . . . . .

33. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ ±. −
. ∞ . . ∞. .
+
f
.(x)
1
. 2
.
So
lim f(x) = +∞ lim f(x) = −∞
x→1− x→2−
lim f(x) = −∞ lim f(x) = +∞
x→1+ x→2+
. . . . . .

34. In English, now
To explain the limit, you can say:
“As x → 1− , the numerator approaches 2, and the denominator
approaches 0 while remaining positive. So the limit is +∞.”
. . . . . .

35. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .

36. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .

37. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .

38. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .

39. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .

40. Limit Laws (?) with infinite limits
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x→a x→a x→a
That is,
∞ .
. +∞=∞
If lim f(x) = −∞ and lim g(x) = −∞, then
x→a x→a
lim (f(x) + g(x)) = −∞. That is,
x→a
− .
. ∞ − ∞ = −∞
. . . . . .

41. Rules of Thumb with infinite limits
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x→a x→a x→a
That is,
∞ .
. +∞=∞
If lim f(x) = −∞ and lim g(x) = −∞, then
x→a x→a
lim (f(x) + g(x)) = −∞. That is,
x→a
− .
. ∞ − ∞ = −∞
. . . . . .

42. Rules of Thumb with infinite limits
If lim f(x) = L and lim g(x) = ±∞, then
x→a x→a
lim (f(x) + g(x)) = ±∞. That is,
x→a
L+∞=∞
. .
L − ∞ = −∞
. . . . . .

43. Rules of Thumb with infinite limits
Kids, don’t try this at home!
The product of a finite limit and an infinite limit is infinite if
the finite limit is not 0.
{
∞. if L > 0
. ·∞=
L
−∞ if L < 0.
{
−∞ if L > 0
. · (−∞) = .
L
∞ if L < 0.
. . . . . .

44. Multiplying infinite limits
Kids, don’t try this at home!
The product of two infinite limits is infinite.
∞·∞=∞
. .
∞ · (−∞) = −∞
(−∞) · (−∞) = ∞
. . . . . .

45. Dividing by Infinity
Kids, don’t try this at home!
The quotient of a finite limit by an infinite limit is zero:
L .
. =0
∞
. . . . . .

46. Dividing by zero is still not allowed
1 .
. =∞
0
There are examples of such limit forms where the limit is ∞, −∞,
undecided between the two, or truly neither.
. . . . . .

47. Indeterminate Limit forms
L
Limits of the form are indeterminate. There is no rule for
0
evaluating such a form; the limit must be examined more closely.
Consider these:
1 −1
lim =∞ lim = −∞
x→0 x2 x→0 x2
1 1
lim
+ x
=∞ lim = −∞
x→0 x→0 − x
1 L
Worst, lim is of the form , but the limit does not
x→0 x sin(1/x) 0
exist, even in the left- or right-hand sense. There are infinitely
many vertical asymptotes arbitrarily close to 0!
. . . . . .

48. Indeterminate Limit forms
Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate.
Example
1
The limit lim sin x · is of the form 0 · ∞, but the answer is
x→0+ x
1.
1
The limit lim sin2 x · is of the form 0 · ∞, but the answer is
x→0+ x
0.
1
The limit lim sin x · 2 is of the form 0 · ∞, but the answer is
x→0 + x
∞.
Limits of indeterminate forms may or may not “exist.” It will
depend on the context.
. . . . . .

49. Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
. . . . . .

50. Outline
Infinite Limits
Vertical Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
. . . . . .

51. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we
like, by taking x sufficiently large.
. . . . . .

52. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we
like, by taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve
y = f(x) if either
lim f(x) = L or lim f(x) = L.
x→∞ x→−∞
. . . . . .

53. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we
like, by taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve
y = f(x) if either
lim f(x) = L or lim f(x) = L.
x→∞ x→−∞
y = L is a horizontal line!
. . . . . .

54. Basic limits at infinity
Theorem
Let n be a positive integer. Then
1
lim =0
x→∞ xn
1
lim n = 0
x→−∞ x
. . . . . .

55. Using the limit laws to compute limits at ∞
Example
Find
2x3 + 3x + 1
lim
x→∞ 4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
. . . . . .

56. Using the limit laws to compute limits at ∞
Example
Find
2x3 + 3x + 1
lim
x→∞ 4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
. . . . . .

57. Solution
Factor out the largest power of x from the numerator and
denominator. We have
2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 )
= 3
4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 )
2x3 + 3x + 1 2 + 3/x2 + 1/x3
lim = lim
x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3
2+0+0 1
= =
4+0+0 2
. . . . . .

58. Solution
Factor out the largest power of x from the numerator and
denominator. We have
2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 )
= 3
4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 )
2x3 + 3x + 1 2 + 3/x2 + 1/x3
lim = lim
x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3
2+0+0 1
= =
4+0+0 2
Upshot
When finding limits of algebraic expressions at infinity, look at
the highest degree terms.
. . . . . .

59. Another Example
Example
x
Find lim
x→∞ x2 +1
. . . . . .

60. Another Example
Example
x
Find lim
x→∞ x2 +1
Answer
The limit is 0.
. . . . . .

61. Solution
Again, factor out the largest power of x from the numerator and
denominator. We have
x x(1) 1 1
= 2 = ·
x2 + 1 x (1 + 1/x2 ) x 1 + 1/x2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2
1
=0· = 0.
1+0
. . . . . .

62. Another Example
Example
x
Find lim
x→∞ x2 +1
Answer
The limit is 0.
y
.
. x
.
. . . . . .

63. Another Example
Example
x
Find lim
x→∞ x2 +1
Answer
The limit is 0.
y
.
. x
.
Notice that the graph does cross the asymptote, which
contradicts one of the heuristic definitions of asymptote.
. . . . . .

64. Solution
Again, factor out the largest power of x from the numerator and
denominator. We have
x x(1) 1 1
= 2 = ·
x2 + 1 x (1 + 1/x2 ) x 1 + 1/x2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2
1
=0· = 0.
1+0
Remark
Had the higher power been in the numerator, the limit would
have been ∞.
. . . . . .

65. Another Example
Example
Find
√
3x4 + 7
lim
x→∞ x2 + 3
. . . . . .

66. Another Example
Example
Find
√ √ √
. 3x4 + 7 ∼ . 3x4 = 3x2
√
3x4 + 7
lim
x→∞ x2 + 3
Answer √
The limit is 3.
. . . . . .

67. Solution
√ √
3x4 + 7 x4 (3 + 7/x4 )
lim = lim 2
x→∞ x2 + 3 x→∞ x (1 + 3/x2 )
√
x2 (3 + 7/x4 )
= lim 2
x→∞ x (1 + 3/x2 )
√
(3 + 7/x4 )
= lim
x→∞ 1 + 3/x2
√
3+0 √
= = 3.
1+0
. . . . . .

68. Rationalizing to get a limit
Example (√ )
Compute lim 4x2 + 17 − 2x .
x→∞
. . . . . .

69. Rationalizing to get a limit
Example (√ )
Compute lim 4x2 + 17 − 2x .
x→∞
Solution
This limit is of the form ∞ − ∞, which we cannot use. So we
rationalize the numerator (the denominator is 1) to get an
expression that we can use the limit laws on.
(√ ) (√ ) √4x2 + 17 + 2x
lim 4x2 + 17 − 2x = lim 4x2 + 17 − 2x · √
x→∞ x→∞ 4x2 + 17 + 2x
(4x 2 + 17) − 4x2
= lim √
x→∞ 4x2 + 17 + 2x
17
= lim √ =0
x→∞ 4x 2 + 17 + 2x
. . . . . .

70. Summary
Infinity is a more complicated concept than a single number.
There are rules of thumb, but there are also exceptions.
Take a two-pronged approach to limits involving infinity:
Look at the expression to guess the limit.
Use limit rules and algebra to verify it.
. . . . . .