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
![### Function and Limit Analysis
Consider the following piecewise function \( f(x) \):
\[
f(x) = \begin{cases}
x^2 & \text{if } x \leq 2 \\
8 - 2x & \text{if } 2 < x < 4 \\
6 & \text{if } x \geq 4
\end{cases}
\]
#### Graphing the Function:
You are required to sketch the given function on the provided graph. Ensure the sketch is exact according to the piecewise definition given.
#### Graph Analysis for Limits:
Using your graph, identify all values of \( c \) for which \( \lim_{x \to c} f(x) \) exists.
**Options:**
a. The limit exists for all points on the graph except where \( c = 2 \).
b. The limit exists for all points on the graph except where \( c = 4 \).
c. The limit exists for all points on the graph except where \( c = 2 \) and \( c = 4 \).
d. The limit exists for all points on the graph.
### Detailed Explanation:
1. **For \( x \leq 2 \)**:
- The function \( f(x) = x^2 \) should be plotted from negative infinity up to and including \( x = 2 \).
2. **For \( 2 < x < 4 \)**:
- The function \( f(x) = 8 - 2x \) should be drawn for values greater than 2 and less than 4. Note that at \( x = 2 \), \( f(x) \) should not be included in this interval, so there should be an open circle at \( x = 2 \).
3. **For \( x \geq 4 \)**:
- The function \( f(x) = 6 \) represents a constant value for \( x \ge 4 \). At \( x = 4 \), this value is exactly 6 and it continues horizontally to positive infinity.
### Using Graph for Limits:
- If you closely examine the graph near **\( x = 2 \)** and **\( x = 4 \)** using the drawn function lines:
- At \( x = 2 \), the left-hand limit \( \lim_{x \to 2^-} f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F880b5248-7c0b-499c-91d5-d57492b6ff41%2F7f2439ad-9523-4587-b96f-6331f634e907%2Fzm9hxl.jpeg&w=3840&q=75)
Transcribed Image Text:### Function and Limit Analysis
Consider the following piecewise function \( f(x) \):
\[
f(x) = \begin{cases}
x^2 & \text{if } x \leq 2 \\
8 - 2x & \text{if } 2 < x < 4 \\
6 & \text{if } x \geq 4
\end{cases}
\]
#### Graphing the Function:
You are required to sketch the given function on the provided graph. Ensure the sketch is exact according to the piecewise definition given.
#### Graph Analysis for Limits:
Using your graph, identify all values of \( c \) for which \( \lim_{x \to c} f(x) \) exists.
**Options:**
a. The limit exists for all points on the graph except where \( c = 2 \).
b. The limit exists for all points on the graph except where \( c = 4 \).
c. The limit exists for all points on the graph except where \( c = 2 \) and \( c = 4 \).
d. The limit exists for all points on the graph.
### Detailed Explanation:
1. **For \( x \leq 2 \)**:
- The function \( f(x) = x^2 \) should be plotted from negative infinity up to and including \( x = 2 \).
2. **For \( 2 < x < 4 \)**:
- The function \( f(x) = 8 - 2x \) should be drawn for values greater than 2 and less than 4. Note that at \( x = 2 \), \( f(x) \) should not be included in this interval, so there should be an open circle at \( x = 2 \).
3. **For \( x \geq 4 \)**:
- The function \( f(x) = 6 \) represents a constant value for \( x \ge 4 \). At \( x = 4 \), this value is exactly 6 and it continues horizontally to positive infinity.
### Using Graph for Limits:
- If you closely examine the graph near **\( x = 2 \)** and **\( x = 4 \)** using the drawn function lines:
- At \( x = 2 \), the left-hand limit \( \lim_{x \to 2^-} f
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