1. Lesson 6
Continuity, Infinite Limits
Math 1a
October 5, 2007
Announcements
No class Monday 10/8, yes class Friday 10/12
MQC closed Sunday, open Monday

2. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f (a) and f (b), where f (a) = f (b). Then
there exists a number c in (a, b) such that f (c) = N.
f (x)
f (b)
N
f (a)
x
a c1 c2 c3 b

3. Using the IVT
Example
Let f (x) = x 3 − x − 1. Show that there is a zero for f . Estimate it
within 1/16.

6. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.

8. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your weight
in pounds.

10. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your weight
in pounds.
True or False
Right now there are two points on opposite sides of the Earth with
exactly the same temperature.

12. 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 (as large as we please) by taking x sufficiently close to a
but not equal to a.
Of course we have definitions for left- and right-hand infinite limits.

14. 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:
limx→a f (x) = ∞ limx→a f (x) = −∞
limx→a+ f (x) = ∞ limx→a+ f (x) = −∞
limx→a− f (x) = ∞ limx→a− f (x) = −∞

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

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

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

18. − +
0
(t − 1)
1

19. − +
0
(t − 1)
1
− +
0
(t − 2)
2

20. − +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)

21. − +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
f (t)
1 2

22. − +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
+
f (t)
1 2

23. − +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
±∞
+
f (t)
1 2

24. − +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
±∞ −
+
f (t)
1 2

25. − +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
±∞ − ∞
+
f (t)
1 2

26. − +
0
(t − 1)
1
− +
0
(t − 2)
2
+
(t 2 + 2)
±∞ − ∞
+ +
f (t)
1 2

28. Limit Laws with infinite limits
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 − ∞ = −∞

29. Rules of Thumb with infinite limits
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 − ∞ = −∞

30. 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.
∞

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

32. 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

33. Example
√
4x 2 + 17 − 2x .
Compute limx→∞

34. Example
√
4x 2 + 17 − 2x .
Compute limx→∞
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.