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Answer the question. Does lim1 f(x) exist? If it does, find the limit. [-x²+1, -1

Answer the question. Does lim1 f(x) exist? If it does, find the limit. [-x²+1, -1

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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Title: Evaluating Limits of a Piecewise Function --- **Question:** Does \(\lim_{x \to 1} f(x)\) exist? If it does, find the limit. **Function Definition:** \[ f(x) = \begin{cases} -x^2 + 1, & \text{for } -1 \leq x < 0 \\ 4x, & \text{for } 0 < x < 1 \\ -5, & \text{for } x = 1 \\ -4x + 8, & \text{for } 1 < x \leq 3 \\ 1, & \text{for } 3 < x < 5 \end{cases} \] **Answer Choices:** - Yes, and it's 4 - No - Yes, and it's -5 - Yes, and it's -4 --- **Explanation:** To determine if the limit exists as \(x\) approaches 1, evaluate the left-hand and right-hand limits, and check the function value at \(x = 1\). If both one-sided limits are equal, the two-sided limit exists. However, the function is defined as -5 at \(x = 1\). By examining the piecewise sections: - As \(x\) approaches 1 from the left (\(0 < x < 1\)), use \(4x\). - As \(x\) approaches 1 from the right (\(1 < x \leq 3\)), use \(-4x + 8\). Compare these results to conclude if the limit exists and identify the correct answer.
Transcribed Image Text:Title: Evaluating Limits of a Piecewise Function --- **Question:** Does \(\lim_{x \to 1} f(x)\) exist? If it does, find the limit. **Function Definition:** \[ f(x) = \begin{cases} -x^2 + 1, & \text{for } -1 \leq x < 0 \\ 4x, & \text{for } 0 < x < 1 \\ -5, & \text{for } x = 1 \\ -4x + 8, & \text{for } 1 < x \leq 3 \\ 1, & \text{for } 3 < x < 5 \end{cases} \] **Answer Choices:** - Yes, and it's 4 - No - Yes, and it's -5 - Yes, and it's -4 --- **Explanation:** To determine if the limit exists as \(x\) approaches 1, evaluate the left-hand and right-hand limits, and check the function value at \(x = 1\). If both one-sided limits are equal, the two-sided limit exists. However, the function is defined as -5 at \(x = 1\). By examining the piecewise sections: - As \(x\) approaches 1 from the left (\(0 < x < 1\)), use \(4x\). - As \(x\) approaches 1 from the right (\(1 < x \leq 3\)), use \(-4x + 8\). Compare these results to conclude if the limit exists and identify the correct answer.
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