1. Copyright © Cengage Learning. All rights reserved.1
Limits and Their Properties
Copyright © Cengage Learning. All rights reserved.

2. Limits involving Infinity
Day 2
BC Limits Review
Limits involving Infinity
Handout: (Introduction to Limits (Graphical)WS)
1.4B-1.5
& 3.5
Copyright © Cengage Learning. All rights reserved.

3. One-Sided Limits and Continuity on a Closed Interval

4. One-Sided Limits and Continuity on a Closed Interval
Figure 1.31

5. Example 4 – Continuity on a Closed Interval
Discuss the continuity of f(x) =
Solution:
The domain of f is the closed interval [–1, 1].

6. Example 4 – Solution Because and you can conclude that f is
cont’d
Because
and
you can conclude that f is
continuous on the closed interval
[–1, 1], as shown in Figure 1.32.
Figure 1.32

7. Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same as the value of the function. 1 2 3 4
This function has discontinuities at x=1 and x=2.
It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

8. Properties of Continuity

9. Properties of Continuity
Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

10. Example – Applying Properties of Continuity
By Theorem 1.11, it follows that each of the functions below is continuous at every point in its domain.

11. The Intermediate Value Theorem
(IVT)

12. The Intermediate Value Theorem
In other words

13. Intermediate Value Theorem
If a function is continuous between a and b, then it takes on every value between and
Because the function is continuous, it must take on every y value between and

14. The Intermediate Value Theorem
The Intermediate Value Theorem tells you that at least one number c exists, but it does not provide a method for finding c. Such theorems are called existence theorems.
The Intermediate Value Theorem states that for a continuous function f, if x takes on all values between a and b, f(x) must take on all values between f(a) and f(b).

15. The Intermediate Value Theorem
Suppose that a girl is 5 feet tall on her thirteenth birthday and 5 feet 7 inches tall on her fourteenth birthday.
Then, for any height h between 5 feet and 5 feet 7 inches, there must have been a time t when her height was exactly h.
This seems reasonable because human growth is continuous and a person’s height does not abruptly change from one value to another.

16. The Intermediate Value Theorem
The Intermediate Value Theorem guarantees the existence of at least one number c in the closed interval [a, b] .
There may, of course, be more than
one number c such that f(c) = k,
as shown in Figure 1.35.
Figure 1.35

17. The Intermediate Value Theorem
A function that is not continuous does not necessarily exhibit the intermediate value property.
For example, the graph of the
function shown in Figure 1.36 jumps
over the horizontal line given by y = k,
and for this function there is no value
of c in [a, b] such that f(c) = k.
Figure 1.36

18. The Intermediate Value Theorem
The Intermediate Value Theorem often can be used to locate the zeros of a function that is continuous on a closed interval.
Specifically, if f is continuous on [a, b] and f(a) and f(b) differ in sign, the Intermediate Value Theorem guarantees the existence of at least one zero of f in the closed interval [a, b] .

19. Note that f is continuous on the closed interval [0, 1]. Because
Example – An Application of the Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the polynomial function has a zero in
the interval [0, 1].
Solution:
Note that f is continuous on the closed interval [0, 1].
Because
it follows that f(x) must pass through zero in the interval [0,2].

20. Solution
cont’d
You can therefore apply the Intermediate Value Theorem to conclude that there must be some c in [0, 1] such that
as shown in Figure 1.37.
Figure 1.37

21. AP Example:
Verify that the Intermediate Value Theorem applies on the interval and find the value of c guaranteed by the theorem.

22. You try an another AP Example:
Explain why the function has a zero in the given interval.
f(x) is continuous
f(0) = -2 and f(1) = 2
Therefore, by the Intermediate Value Theorem (IVT), f(x)=0 for at least one value of c between 0 and 1.

23. Discuss with your neighbor
Name the three conditions that must be met for a function to be continuous at a point.

24. 1.5
Infinite Limits

25. Infinite Limits:
As the denominator approaches zero, the value of the fraction gets very large.
vertical asymptote at x=0.
If the denominator is positive then the fraction is positive.
If the denominator is negative then the fraction is negative.

26. Limits at infinity:
vertical asymptote at x=0.
What happens to the value of this expression as the denominator approaches either positive or negative infinity?

27. Example – Determining Infinite Limits from a Graph
Determine the limit of each function shown in Figure 1.41 as x approaches 1 from the left and from the right.
Figure 1.41

28. Example 1(a) – Solution
When x approaches 1 from the left or the right,
(x – 1)2 is a small positive number.
Thus, the quotient 1/(x – 1)2 is a large positive number and f(x) approaches infinity from each side of x = 1.
So, you can conclude that
Figure 1.41(a) confirms this analysis.
Figure 1.41(a)

29. Example 1(b) – Solution
cont’d
When x approaches 1 from the left, x – 1 is a small negative number.
Thus, the quotient –1/(x – 1) is a large positive number and f(x) approaches infinity from left of x = 1.
So, you can conclude that
When x approaches 1 from the right, x – 1 is a small positive number.

30. Example 1(b) – Solution
cont’d
Thus, the quotient –1/(x – 1) is a large negative number and f(x) approaches negative infinity from the right of x = 1.
So, you can conclude that
Figure 1.41(b) confirms this analysis.
What is the limit of this function at x=1?
DNE
Figure 1.41(b)

31. Example 2 – Finding Vertical Asymptotes
Determine all vertical asymptotes of the graph of each function.

32. Example 2(a) – Solution
When x = –1, the denominator of is 0 and the numerator is not 0.
So, by Theorem 1.14, you can conclude that x = –1 is a vertical asymptote, as shown in Figure 1.42(a).
Figure 1.42(a).

33. Example 2(b) – Solution By factoring the denominator as
cont’d
By factoring the denominator as
you can see that the denominator is 0 at x = –1 and x = 1.
Moreover, because the numerator is not 0 at these two points, you can apply Theorem 1.14 to conclude that the graph of f has two vertical asymptotes, as shown in figure 1.42(b).
Figure 1.42(b)

34. Example 2(c) – Solution By writing the cotangent function in the form
cont’d
By writing the cotangent function in the form
you can apply Theorem 1.14 to
conclude that vertical asymptotes
occur at all values of x such that
sin x = 0 and cos x ≠ 0, as shown
in Figure 1.42(c).
So, the graph of this function has infinitely many vertical asymptotes. These asymptotes occur at x = nπ, where n is an integer.
Figure 1.42(c).

35. You try:

37. You try:
Find the limit: a) b) c)

38. You try:

39. Limits at Infinity
This section discusses the “end behavior” of a function
on an infinite interval. Consider the graph of
as shown in Figure 3.33.
Figure 3.33

40. Limits at Infinity
Graphically, you can see that the values of f(x) appear to approach 3 as x increases without bound or decreases without bound. You can come to the same conclusions numerically, as shown in the table.

41. Limits at Infinity
The table suggests that the value of f(x) approaches 3 as x increases without bound Similarly, f(x) approaches 3 as x decreases without bound
These limits at infinity are denoted by
and

42. Horizontal Asymptotes
Limits at infinity are horizontal asymptotes.
For rational functions, use horizontal asymptote rules.

43. Example – Finding a Limit at Infinity
This number becomes insignificant as
There is a horizontal asymptote at 1.

44. Same Example – Algebraic Solution
There is a horizontal asymptote at 1.

45. Find:
Example:
When we graph this function, the limit appears to be zero.
so for :
by the sandwich theorem:

46. Example:
Find:

47. Example – Finding a Limit at Infinity
Find the limit:
Solution:
Using Theorem 3.10, you can write

48. Example – Finding a Limit at Infinity
Find the limit:
Solution:

49. Example – Finding a Limit at Infinity

50. Example – Finding a Limit at Infinity

51. Example – A Function with Two Horizontal Asymptotes
Find each limit.

52. Example 4 – Solution The graph of is shown in figure 3.38. cont’d

53. Often you can just “think through” limits.p

54. Example – Finding Infinite Limits at Infinity
Find each limit.
Solution:
As x increases without bound, x3 also increases without bound. So, you can write
As x decreases without bound, x3 also decreases without bound. So, you can write

55. Example – Solution
cont’d
The graph of f(x) = x3 in Figure 3.42 illustrates these two
results. These results agree with the Leading Coefficient Test for polynomial functions.
Figure 3.42

56. Discuss with your group

57. You are in Calculus, therefore you rock!
Theorem 1.1
You are in Calculus, therefore you rock!

58. Find the following limit:
HWQ
Find the following limit:

59. Homework
MMM pgs
+ 76,77,83,84 pg. 80 Larson

60. Find the following limit:
HWQ
Find the following limit:

61. Find the following limit:
HWQ
Find the following limit: