Refer to the diagram on the right. Note the open circles. 1. lim f(x) = x-3- 2. lim f(x)= x→3+ = 3. lim f(x): x →3 4. lim f(x): = x→ 1 5. f(1) = = 2 0 2 3
Refer to the diagram on the right. Note the open circles. 1. lim f(x) = x-3- 2. lim f(x)= x→3+ = 3. lim f(x): x →3 4. lim f(x): = x→ 1 5. f(1) = = 2 0 2 3
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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![### Problem 2
**\( f(x) \) is a piecewise function defined as:**
\[
f(x) =
\begin{cases}
x + 1 & -2 < x < 0 \\
2 & x = 0 \\
-x & 0 < x < 2 \\
0 & x = 2 \\
x - 4 & 2 < x \leq 4
\end{cases}
\]
**The graph of the function is provided below. Answer the following questions.**
**Description of the Graph:**
- The graph is plotted on a Cartesian plane.
- There are five segments corresponding to the different cases in the piecewise function.
- For \( -2 < x < 0 \), the graph is a linear line increasing from \(-1\) to \(1\) at \( x = 0 \).
- At \( x = 0 \), there is a solid dot at \( (0, 2) \), indicating \( f(0) = 2 \).
- For \( 0 < x < 2 \), the graph is a line decreasing from \( 2.4 \) to \( 0 \).
- At \( x = 2 \), there is an open circle at \( (2, 0) \) and a solid dot at \( (2, -2) \), signifying the value of the function jumps at \( x = 2 \).
- For \( 2 < x \leq 4 \), the graph is a line increasing from \( (2, 2) \), \( (-2) \) to \( (4, 0) \).
**Questions:**
1. \(\displaystyle \lim_{{x \to -2^+}} f(x) =\)
2. \(\displaystyle \lim_{{x \to 0^-}} f(x) =\)
3. \(\displaystyle \lim_{{x \to 0^+}} f(x) =\)
4. \( f(0) =\)
5. \(\displaystyle \lim_{{x \to 2^-}} f(x) =\)
6. \(\displaystyle \lim_{{x \to 2^+}} f(x) =\)
7. \( f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30f2ef89-ce64-4161-b7cf-3fd72303d16e%2F35fd429e-8b8a-42d2-b40c-d2f61e821cd0%2F1dogjgg_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2
**\( f(x) \) is a piecewise function defined as:**
\[
f(x) =
\begin{cases}
x + 1 & -2 < x < 0 \\
2 & x = 0 \\
-x & 0 < x < 2 \\
0 & x = 2 \\
x - 4 & 2 < x \leq 4
\end{cases}
\]
**The graph of the function is provided below. Answer the following questions.**
**Description of the Graph:**
- The graph is plotted on a Cartesian plane.
- There are five segments corresponding to the different cases in the piecewise function.
- For \( -2 < x < 0 \), the graph is a linear line increasing from \(-1\) to \(1\) at \( x = 0 \).
- At \( x = 0 \), there is a solid dot at \( (0, 2) \), indicating \( f(0) = 2 \).
- For \( 0 < x < 2 \), the graph is a line decreasing from \( 2.4 \) to \( 0 \).
- At \( x = 2 \), there is an open circle at \( (2, 0) \) and a solid dot at \( (2, -2) \), signifying the value of the function jumps at \( x = 2 \).
- For \( 2 < x \leq 4 \), the graph is a line increasing from \( (2, 2) \), \( (-2) \) to \( (4, 0) \).
**Questions:**
1. \(\displaystyle \lim_{{x \to -2^+}} f(x) =\)
2. \(\displaystyle \lim_{{x \to 0^-}} f(x) =\)
3. \(\displaystyle \lim_{{x \to 0^+}} f(x) =\)
4. \( f(0) =\)
5. \(\displaystyle \lim_{{x \to 2^-}} f(x) =\)
6. \(\displaystyle \lim_{{x \to 2^+}} f(x) =\)
7. \( f
Transcribed Image Text:### Problem 1
**Refer to the diagram on the right. Note the open circles.**
1. \(\lim\limits_{x \to 3^-} f(x) =\)
2. \(\lim\limits_{x \to 3^+} f(x) =\)
3. \(\lim\limits_{x \to 3} f(x) =\)
4. \(\lim\limits_{x \to 1} f(x) =\)
5. \(f(1) =\)
**Explanation of the Graph:**
The graph illustrates a piecewise function represented by a series of connected lines with open and closed circles indicating specific values. The \(x\)-axis ranges from 0 to 5, while the \(y\)-axis ranges from 0 to 3.
- The graph starts at point \((0, 1)\) with a closed circle.
- It increases linearly to \((1, 2)\) with an open circle.
- Then it moves horizontally to \((2, 2)\) with a solid line and ends with a closed circle at \((2, 2)\).
- From \((2, 2)\) it decreases to \((3, 1)\) ending with a solid circle.
- It continues decreasing linearly towards \((4, 0)\) with the open circle.
#### Understanding the Open and Closed Circles:
- An open circle indicates that the function does not include that endpoint.
- A closed circle indicates that the function includes that endpoint.
Use these interpretations to answer the limits and function values for the given points.
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