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...
Related questions
Question
6)
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb034729f-db63-4235-9b33-79c3c77be50a%2Fdd7dea7f-b502-48fb-8ea7-48a3568c8b81%2Fg2srzqc_processed.jpeg&w=3840&q=75)
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.
Expert Solution
Step 1
To find the limit of the piecewise function
Step by step
Solved in 3 steps with 1 images
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