1. Splash Screen

2. Five-Minute Check (over Chapter 11) Then/Now New Vocabulary
Example 1: Estimate a Limit = f (c)
Example 2: Estimate a Limit ≠ f (c)
Key Concept: Independence of Limit from Function Value at a Point
Key Concept: One-Sided Limits
Key Concept: Existence of a Limit at a Point
Example 3: Estimate One-Sided and Two-Sided Limits
Example 4: Limits and Unbounded Behavior
Example 5: Limits and Oscillating Behavior
Concept Summary: Why Limits at a Point Do Not Exist
Key Concept: Limits at Infinity
Example 6: Estimate Limits at Infinity
Example 7: Real-World Example: Estimate Limits at Infinity
Lesson Menu

3. A. discrete; the number of baseball players is countable.
Classify the random variable X, where X represents the number of players on a baseball team, as discrete or continuous. Explain your reasoning.
A. discrete; the number of baseball players is countable.
B. continuous; the number of baseball players is countable.
C. discrete; the number of players on a baseball team varies from team to team.
D. continuous; the number of players on a baseball team can be anywhere from 9 to 24 players.
5–Minute Check 1

4. Find X if z = 2.91, μ = 49, and σ = 2.7. A B C D
5–Minute Check 2

5. Workers at a candle company test one type of candle and find that 77% burn for more than 14 hours before extinguishing. What is the probability that in a random sample of 35 candles, more than 30 will burn for more than 14 hours?
A. 7.0%
B. 15.2%
C. 84.8%
D. 93.0%
5–Minute Check 3

6. A random survey of 40 people in a grocery store showed that the average time spent shopping was 18.7 minutes. Assume the standard deviation from a previous survey was 1.4 minutes. Which of the following represents the maximum error of estimate given a 95% confidence level?
A minute
B minute
C minute
D minute
5–Minute Check 4

7. Estimate limits of functions at fixed values.
You estimated limits to determine the continuity and end behavior of functions. (Lesson 1-3)
Estimate limits of functions at fixed values.
Estimate limits of functions at infinity.
Then/Now

8. one-sided limit
two-sided limit
Vocabulary

9. Estimate a Limit = f (c)
Estimate using a graph. Support your conjecture using a table of values.
Analyze Graphically
The graph of f (x) = 4x + 1 suggests that as x gets closer to –7, the corresponding function values get closer to –27. Therefore, we can estimate that = –27.
Example 1

10. Estimate a Limit = f (c)
Support Numerically
Make a table of values for f, choosing x-values that approach –7 by using some values slightly less than –7 and some values slightly greater than –7.
Example 1

11. Estimate a Limit = f (c)
The pattern of outputs suggests that as x gets close to –7 from the left or right, f (x) gets closer to –27. This supports our graphical analysis.
Answer: –27
Example 1

12. Estimate using a graph.
A. 3,
B. 1,
C. –1,
D. –3,
Example 1

13. Estimate a Limit ≠ f (c)
Estimate using a graph. Support your conjecture using a table of values.
Analyze Graphically
The graph of suggests that as x gets closer to 4, the corresponding function value approaches 8. Therefore, we can estimate that is 8.
Example 2

14. Estimate a Limit ≠ f (c)
Support Numerically
Make a table of values, choosing x-values that approach 4 from either side.
Example 2

15. Estimate a Limit ≠ f (c)
The pattern of outputs suggests that as x gets closer to 4, f (x) gets closer to 8. This supports our graphical analysis.
Answer: 8
Example 2

16. Estimate using a graph.
A. 0,
B. 0,
C. 6,
D. –6,
Example 2

17. Key Concept 3

18. Key Concept 3

19. Key Concept 3

20. A. Estimate each one-sided or two-sided limit, if it exists.
Estimate One-Sided and Two-Sided Limits
A. Estimate each one-sided or two-sided limit, if it exists.
The graph of suggests that f (x) = –2 and f (x) = 3. Because the left- and right-hand limits of f (x) as x approaches 1 are not the same, does not exist.
Example 3

21. Estimate One-Sided and Two-Sided Limits
Answer:
Example 3

22. B. Estimate each one-sided or two-sided limit, if it exists.
Estimate One-Sided and Two-Sided Limits
B. Estimate each one-sided or two-sided limit, if it exists.
The graph of g(x) suggests that g(x) = –1 and g(x) = –1. Because the left- and right-hand limits of g(x) as x approaches 0 are the same, exists and is –1.
Example 3

23. Estimate One-Sided and Two-Sided Limits
Answer:
Example 3

24. Estimate each one-sided or two-sided limit, if it exists.B. C. D.
Example 3

25. A. Estimate , if it exists. Analyze Graphically
Limits and Unbounded Behavior
A. Estimate , if it exists.
Analyze Graphically
The graph of suggests that and because as x gets closer to 2, the function values of the graph increase.
Example 4

26. Limits and Unbounded Behavior
Neither one-sided limit at x = 2 exists; therefore, we can conclude that does not exist. However, because both sides approach ∞, we describe the behavior of f(x) at 2 by writing
Support Numerically
Example 4

27. Limits and Unbounded Behavior
The pattern of outputs suggests that as x gets closer to 2 from the left and the right, f(x) grows without bound. This supports our graphical analysis.
Answer: ∞
Example 4

28. B. Estimate , if it exists. Analyze Graphically
Limits and Unbounded Behavior
B. Estimate , if it exists.
Analyze Graphically
The graph of suggests that and because as x gets closer to 0, the function values from the left decrease and the function values from the right increase.
Example 4

29. Limits and Unbounded Behavior
Neither one-sided limit at x = 0 exists; therefore, does not exist. In this case, we cannot describe the behavior of f(x) at 0 using a single expression because the unbounded behaviors from the left and right differ.
Support Numerically
Example 4

30. Limits and Unbounded Behavior
The pattern of outputs suggests that as x gets closer to 0 from the left and the right, f(x) decreases and increases without bound, respectively. This supports our graphical analysis.
Answer: does not exist
Example 4

31. Use a graph to estimate , if it exists.
B. –∞,
C. ∞,
D. –∞,
Example 4

32. Limits and Oscillating Behavior
Estimate , if it exists.
The graph of f(x) = x sin x suggests that as x gets closer to 0, the corresponding function values get closer and closer to 0.
Example 5

33. Limits and Oscillating Behavior
Therefore,
Answer: 0
Example 5

34. Estimate , if it exists.
A. does not exist
B. 1
C. 0
D. –1
Example 5

35. Concept Summary 6

36. Key Concept 6

37. The graph of suggests that . As x increases, f(x) gets closer to 1.
Estimate Limits at Infinity
A. Estimate , if it exists.
Analyze Graphically
The graph of suggests that As x increases, f(x) gets closer to 1.
Example 6

38. Estimate Limits at Infinity
Support Numerically
The pattern of outputs suggests that as x increases, f(x) approaches 1.
Answer: 1
Example 6

39. The graph of suggests that = –1. As x increases, f(x) gets
Estimate Limits at Infinity
B. Estimate , if it exists.
Analyze Graphically
The graph of suggests that = –1. As x increases, f(x) gets
closer to –1.
Example 6

40. Estimate Limits at Infinity
Support Numerically
The pattern of outputs suggests that as x increases, f(x) approaches –1.
Answer: –1
Example 6

41. C. Estimate , if it exists. Analyze Graphically
Estimate Limits at Infinity
C. Estimate , if it exists.
Analyze Graphically
The graph of f(x) = cos x suggests that does not exist. As x increases, f(x) oscillates between 1 and –1.
Example 6

42. Estimate Limits at Infinity
Support Numerically
The pattern of outputs suggests that as x increases, f(x) oscillates between 1 and –1.
Answer: does not exist
Example 6

43. Estimate , if it exists.
A. –2
B. 2
C. –∞
D. ∞
Example 6

44. Estimate Limits at Infinity
A. BACTERIA The growth of a certain bacteria can be modeled by the logistic growth function , where t represents time in hours. Estimate , if it exists, and interpret your result.
Example 7

45. Estimate Limits at Infinity
Graph using a graphing calculator. The graph shows that when t = 20, B(t) ≈ Notice that as t increases, the function values of the graph get closer and closer to 675. So we can estimate that
Example 7

46. Estimate Limits at Infinity
Interpret the Result
Over time, the population of the bacteria will approach a maximum of 675.
Answer: ; Over time, the population of the bacteria will approach a maximum of 675.
Example 7

47. Estimate Limits at Infinity
B. POPULATION The population growth of a certain city is given by the function P(t) = 0.7(1.1)t, where t is time in years. Estimate , if it exists, and interpret your result.
Graph the function P(t ) = 0.7(1.1)t using a graphing calculator. The graph shows that as t increases the function values increase. So, we can estimate that
Example 7

48. Estimate Limits at Infinity
Interpret the Result
If the pattern continues, the population will grow without bound over time.
Answer: ; If the pattern continues, the population will grow without bound over time.
Example 7

49. A. ; Over time, the deer population will grow without bound.
POPULATION The population growth of deer on Fawn Island is given by P (t) = 200(0.81)t, where t is time given in years. Estimate , if it exists, and interpret your results.
A ; Over time, the deer population will grow without bound.
B ; Over time, the deer population will reach 0.
C ; Over time, the deer population will reach 200.
D ; Over time, the deer population will reach 162.
Example 7

50. End of the Lesson