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11 if x > 8 7 f(x) = if x = 8 if – 9

11 if x > 8 7 f(x) = if x = 8 if – 9

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The image contains a piecewise function \( f(x) \) and a set of limit problems related to it. ### Piecewise Function Definition The function \( f(x) \) is defined as follows: - \( f(x) = 11 \) if \( x > 8 \) - \( f(x) = 7 \) if \( x = 8 \) - \( f(x) = -x + 11 \) if \( -9 \leq x < 8 \) - \( f(x) = 20 \) if \( x < -9 \) ### Limit Problems The task is to find the following limits if they exist. If a limit does not exist, enter "n". 1. \( \lim_{x \to 8^-} f(x) \) - Answer: 7 2. \( \lim_{x \to 8^+} f(x) \) - Answer: 11 3. \( \lim_{x \to 8} f(x) \) - Answer: n 4. \( \lim_{x \to -9} f(x) \) - Answer: 20 5. \( \lim_{x \to -9^+} f(x) \) - Answer: 20 6. \( \lim_{x \to -9^-} f(x) \) - Answer: 20 ### Explanation - For \( x \to 8^- \), \( f(x) \) approaches \( 7 \) as \( x \) comes from the left (using definition: \(-x + 11\)). - For \( x \to 8^+ \), \( f(x) \) approaches \( 11 \) as \( x \) comes from the right. - \( \lim_{x \to 8} f(x) \) does not exist because the left-hand limit (7) and the right-hand limit (11) do not agree. - For \( x \to -9 \), \( x \to -9^+ \), and \( x \to -9^- \), \( f(x) \) consistently approaches \( 20 \). This piecewise function explores the behavior, continuity, and discontinuity in limits, especially around critical points specified by the function's definition.
Transcribed Image Text:The image contains a piecewise function \( f(x) \) and a set of limit problems related to it. ### Piecewise Function Definition The function \( f(x) \) is defined as follows: - \( f(x) = 11 \) if \( x > 8 \) - \( f(x) = 7 \) if \( x = 8 \) - \( f(x) = -x + 11 \) if \( -9 \leq x < 8 \) - \( f(x) = 20 \) if \( x < -9 \) ### Limit Problems The task is to find the following limits if they exist. If a limit does not exist, enter "n". 1. \( \lim_{x \to 8^-} f(x) \) - Answer: 7 2. \( \lim_{x \to 8^+} f(x) \) - Answer: 11 3. \( \lim_{x \to 8} f(x) \) - Answer: n 4. \( \lim_{x \to -9} f(x) \) - Answer: 20 5. \( \lim_{x \to -9^+} f(x) \) - Answer: 20 6. \( \lim_{x \to -9^-} f(x) \) - Answer: 20 ### Explanation - For \( x \to 8^- \), \( f(x) \) approaches \( 7 \) as \( x \) comes from the left (using definition: \(-x + 11\)). - For \( x \to 8^+ \), \( f(x) \) approaches \( 11 \) as \( x \) comes from the right. - \( \lim_{x \to 8} f(x) \) does not exist because the left-hand limit (7) and the right-hand limit (11) do not agree. - For \( x \to -9 \), \( x \to -9^+ \), and \( x \to -9^- \), \( f(x) \) consistently approaches \( 20 \). This piecewise function explores the behavior, continuity, and discontinuity in limits, especially around critical points specified by the function's definition.
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