We examine two ways of extending the definition of limit: A function can be said to have a limit of infinity (or minus infinity) at a point if it grows without bound near that point. A function can have a limit at a point if values of the function get close to a value as the points get arbitrarily large.

1. Section 2.5
Limits Involving Infinity
Math 1a
February 4, 2008
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Syllabus available on course website
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No class Monday 2/18
ALEKS due Wednesday 2/20

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

3. Infinite Limits
Definition
The notation
lim f (x) = ∞
x→a
means that the 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.
Definition
The notation
lim f (x) = −∞
x→a
means that the values of f (x) can be made arbitrarily large
negative by taking x sufficiently close to a but not equal to a.
Of course we have definitions for left- and right-hand infinite limits.

4. 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−

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

6. 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.

7. 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.

8. − 0 +
(t − 1)
1

9. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2

10. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2
+
(t 2 + 2)

11. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2
+
(t 2 + 2)
f (t)
1 2

12. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2
+
(t 2 + 2)
+
f (t)
1 2

13. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2
+
(t 2 + 2)
+ ±∞
f (t)
1 2

14. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2
+
(t 2 + 2)
+ ±∞ −
f (t)
1 2

15. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2
+
(t 2 + 2)
+ ±∞ − ∞
f (t)
1 2

16. − 0 +
(t − 1)
1
− 0 +
(t − 2)
2
+
(t 2 + 2)
+ ±∞ − ∞ +
f (t)
1 2

17. Limit Laws with infinite limits
To aid your intuition
The sum of positive infinite limits is ∞. That is
∞+∞=∞
The sum of negative infinite limits is −∞.
−∞ − ∞ = −∞
The sum of a finite limit and an infinite limit is infinite.
a+∞=∞
a − ∞ = −∞

18. Rules of Thumb with infinite limits
Don’t try this at home!
The sum of positive infinite limits is ∞. That is
∞+∞=∞
The sum of negative infinite limits is −∞.
−∞ − ∞ = −∞
The sum of a finite limit and an infinite limit is infinite.
a+∞=∞
a − ∞ = −∞

19. Rules of Thumb with infinite limits
The product of a finite limit and an infinite limit is infinite if
the finite limit is not 0.
∞ if a > 0
a·∞=
−∞ if a < 0.
−∞ if a > 0
a · (−∞) =
∞ if a < 0.
The product of two infinite limits is infinite.
∞·∞=∞
∞ · (−∞) = −∞
(−∞) · (−∞) = ∞
The quotient of a finite limit by an infinite limit is zero:
a
= 0.
∞

20. 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.

21. 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.
1
Limits of the form are also indeterminate.
0

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

23. 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.

24. 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→−∞

25. 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!

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

27. 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
C 0
D ∞

28. 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
C 0
D ∞

29. 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

30. 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.

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

32. Another Example
Example
Find √
3x 4 + 7
lim
x→∞ x2 + 3
Solution √
The limit is 3.

33. Example
x2
Make a conjecture about lim .
x→∞ 2x

34. Example
x2
Make a conjecture about lim .
x→∞ 2x
Solution
The limit is zero. Exponential growth is infinitely faster than
geometric growth

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

36. Rationalizing to get a limit
Example
Compute lim 4x 2 + 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.

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

38. Worksheet