This document discusses limits involving infinity in calculus. It begins with definitions of infinite limits, both positive and negative infinity. It provides examples of common infinite limits, such as the limit of 1/x as x approaches 0. It also discusses vertical asymptotes and rules for infinite limits, such as the limit of a sum being infinity if both terms have infinite limits. The document contains examples evaluating limits at points where functions are not continuous.

1. Section 1.6
Limits Involving Infinity
V63.0121, Calculus I
February 4–5, 2009
Announcements
Problem Set 2 due today

2. Recall the unboundedness problem
1
Recall why lim+ doesn’t exist.
x
x→0
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.

3. Recall the unboundedness problem
1
Recall why lim+ doesn’t exist.
x
x→0
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
x→0
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
x→0
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. Outline
Infinite Limits
Vertical Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limits
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit

7. 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

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
“Large” takes the place
of “close to L”.

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.
x
“Large” takes the place
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.
x
“Large” takes the place
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.
x
“Large” takes the place
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.
x
“Large” takes the place
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.
x
“Large” takes the place
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.
x
“Large” takes the place
of “close to L”.

15. 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.

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.
We call a number large or small based on its absolute value.
So −1, 000, 000 is a large (negative) number.

17. 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−

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

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

20. Infinite Limits we Know
y
1
=∞
lim+
x→0 x
1
= −∞
lim
−x
x
x→0
1
=∞
lim
x→0 x 2

21. Finding limits at trouble spots
Example
Let
t2 + 2
f (t) =
t 2 − 3t + 2
Find lim f (t) and lim+ f (t) for each a at which f is not
t→a− t→a
continuous.

22. Finding limits at trouble spots
Example
Let
t2 + 2
f (t) =
t 2 − 3t + 2
Find lim f (t) and lim+ f (t) for each a at which f is not
t→a− t→a
continuous.
Solution
The denominator factors as (t − 1)(t − 2). We can record the
signs of the factors on the number line.

23. Use the number line
− +
0
(t − 1)
1

24. Use the number line
− +
0
(t − 1)
1
− +
0
(t − 2)
2

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

26. Use the number line
− +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
f (t)
1 2

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

28. Use the number line
− +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
±∞
+
f (t)
1 2

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

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

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

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

33. The graph so far
y
x
−1 1 2 3

34. The graph so far
y
x
−1 1 2 3

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

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

40. 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 − ∞ = −∞

41. 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.

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

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

44. 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.

45. Indeterminate Limits
Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There
is no rule for evaluating such a form; the limit must be
examined more closely.

46. Indeterminate Limits
Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There
is no rule for evaluating such a form; the limit must be
examined more closely.
L
Limits of the form are also indeterminate. Consider these:
0
−1
1
=∞ = −∞
lim lim
x→0 x 2 x→0 x 2
1 1
lim+ = ∞ = −∞
lim
x→0 x −x
x→0
1
Worst, lim does not exist, even in the left- or
x→0 x sin(1/x)
right-hand sense.

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

48. 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.

49. 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→−∞

50. 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!

51. Theorem
Let n be a positive integer. Then
1
lim n = 0
x→∞ x
1
lim =0
x→−∞ x n

52. Using the limit laws to compute limits at ∞
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 7
if it exists.
A does not exist
B 1/2
C0
D∞

53. Using the limit laws to compute limits at ∞
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 7
if it exists.
A does not exist
B 1/2
C0
D∞

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

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

56. Another Example
Example
Find
√
3x 4 + 7
lim
x2 + 3
x→∞

57. Another Example
Example
Find
√ √
3x 2
3x 4 + 7 ∼ 3x 4 =
√
3x 4 + 7
lim
x2 + 3
x→∞

58. Another Example
Example
Find
√ √
3x 2
3x 4 + 7 ∼ 3x 4 =
√
3x 4 + 7
lim
x2 + 3
x→∞
Answer √
The limit is 3.

59. Solution
√
x 4 (3 + 7/x 4 )
3x 4 + 7
lim = lim
x2 + 3 x→∞ x 2 (1 + 3/x 2 )
x→∞
x 2 (3 + 7/x 4 )
= lim
x→∞ x 2 (1 + 3/x 2 )
(3 + 7/x 4 )
= lim
1 + 3/x 2
x→∞
√
3+0 √
= = 3.
1+0

60. Rationalizing to get a limit
Example
4x 2 + 17 − 2x .
Compute lim
x→∞

61. Rationalizing to get a limit
Example
4x 2 + 17 − 2x .
Compute lim
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.
√
4x 2 + 17 + 2x
4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √
lim
4x 2 + 17 + 2x
x→∞ x→∞
2 + 17) − 4x 2
(4x
= lim √
4x 2 + 17 + 2x
x→∞
17
= lim √ =0
4x 2 + 17 + 2x
x→∞

62. 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.