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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Question
![### Evaluating the Limit of a Piecewise Function as x Approaches 1
Given the following piecewise function, evaluate \(\lim_{{x \to 1^-}} f(x)\).
\[
f(x) =
\begin{cases}
-2x^2 - 2x & \text{if } x \leq -2 \\
3x - 2 & \text{if } -2 < x \leq 1 \\
x^2 + 3 & \text{if } x > 1
\end{cases}
\]
Here's the breakdown of the piecewise function \(f(x)\) based on different intervals of \(x\):
- **For \( x \leq -2 \)**:
\[
f(x) = -2x^2 - 2x
\]
- **For \( -2 < x \leq 1 \)**:
\[
f(x) = 3x - 2
\]
- **For \( x > 1 \)**:
\[
f(x) = x^2 + 3
\]
To evaluate \(\lim_{{x \to 1^-}} f(x)\), we consider the value of \(f(x)\) as \(x\) approaches 1 from the left side (i.e., \(x < 1\)).
For \(-2 < x \leq 1\):
\[
f(x) = 3x - 2
\]
As \(x\) approaches 1 from the left (\(x \to 1^-\)):
\[
f(1^-) = 3(1) - 2 = 3 - 2 = 1
\]
Therefore:
\[
\lim_{{x \to 1^-}} f(x) = 1
\]
No graphs or diagrams are provided in the image, so there are no additional details to explain visually.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb746646f-39dd-4312-845d-6abd1eeee77e%2F6daa8240-6cac-4920-8234-4626c3a057e3%2Ff9k5e2q_processed.png&w=3840&q=75)
Transcribed Image Text:### Evaluating the Limit of a Piecewise Function as x Approaches 1
Given the following piecewise function, evaluate \(\lim_{{x \to 1^-}} f(x)\).
\[
f(x) =
\begin{cases}
-2x^2 - 2x & \text{if } x \leq -2 \\
3x - 2 & \text{if } -2 < x \leq 1 \\
x^2 + 3 & \text{if } x > 1
\end{cases}
\]
Here's the breakdown of the piecewise function \(f(x)\) based on different intervals of \(x\):
- **For \( x \leq -2 \)**:
\[
f(x) = -2x^2 - 2x
\]
- **For \( -2 < x \leq 1 \)**:
\[
f(x) = 3x - 2
\]
- **For \( x > 1 \)**:
\[
f(x) = x^2 + 3
\]
To evaluate \(\lim_{{x \to 1^-}} f(x)\), we consider the value of \(f(x)\) as \(x\) approaches 1 from the left side (i.e., \(x < 1\)).
For \(-2 < x \leq 1\):
\[
f(x) = 3x - 2
\]
As \(x\) approaches 1 from the left (\(x \to 1^-\)):
\[
f(1^-) = 3(1) - 2 = 3 - 2 = 1
\]
Therefore:
\[
\lim_{{x \to 1^-}} f(x) = 1
\]
No graphs or diagrams are provided in the image, so there are no additional details to explain visually.
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