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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

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

Find X if z = 2.91, μ = 49, and σ = 2.7. A. 145.29 B. 135.21 C. 56.86 D. 54.61 5–Minute Check 2

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

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. 0.36 minute B. 0.43 minute C. 0.45 minute D. 0.57 minute 5–Minute Check 4

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

one-sided limit two-sided limit Vocabulary

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

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

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

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

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

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

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

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

Key Concept 3

Key Concept 3

Key Concept 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Concept Summary 6

Key Concept 6

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

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

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

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

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

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

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

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

Estimate Limits at Infinity Graph using a graphing calculator. The graph shows that when t = 20, B(t) ≈ 674.44. Notice that as t increases, the function values of the graph get closer and closer to 675. So we can estimate that . Example 7

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

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

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

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

End of the Lesson