Let ーX if x < -1 g(x) = {1 - x2 if -1 < x < 1. х- 1 if x > 1 (a) Evaluate each of the following limits, if it exists. (If an answer does not exist, enter DNE.) (i) lim g(x) x → 1+ (ii) lim g(x) x 1 (iii) lim im, g(x) (iv) lim g(x) x → -1 (v) lim g(x) x → -1+ (vi) lim g(x) x -1 (b) Sketch the graph of g. g(x) go 3 3
Let ーX if x < -1 g(x) = {1 - x2 if -1 < x < 1. х- 1 if x > 1 (a) Evaluate each of the following limits, if it exists. (If an answer does not exist, enter DNE.) (i) lim g(x) x → 1+ (ii) lim g(x) x 1 (iii) lim im, g(x) (iv) lim g(x) x → -1 (v) lim g(x) x → -1+ (vi) lim g(x) x -1 (b) Sketch the graph of g. g(x) go 3 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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![## Function and Limits Analysis
### Given Piecewise Function:
Let
\[
g(x) =
\begin{cases}
-x & \text{if } x \leq -1 \\
1 - x^2 & \text{if } -1 < x < 1 \\
x - 1 & \text{if } x > 1
\end{cases}
\]
### (a) Evaluate the following limits, if they exist. (If an answer does not exist, enter DNE.)
1. (i) \( \lim_{x \to 1^+} g(x) \)
[Input Box]
2. (ii) \( \lim_{x \to 1} g(x) \)
[Input Box]
3. (iii) \( \lim_{x \to 0} g(x) \)
[Input Box]
4. (iv) \( \lim_{x \to -1} g(x) \)
[Input Box]
5. (v) \( \lim_{x \to -1^+} g(x) \)
[Input Box]
6. (vi) \( \lim_{x \to -1^-} g(x) \)
[Input Box]
### (b) Sketch the graph of \( g \).
### Graph Explanation:
The graph includes three distinct parts based on the piecewise function definition:
- **For \( x \leq -1 \)**: The section of the graph represents the linear function \( -x \). It is a straight line with a negative slope, leading downwards as \( x \) decreases.
- **For \( -1 < x < 1 \)**: This part displays the quadratic function \( 1 - x^2 \). The graph within this interval is a downward-opening parabola with its vertex at \( (0, 1) \).
- **For \( x > 1 \)**: Here, the graph corresponds to the linear function \( x - 1 \), which is a line with a positive slope, moving upwards as \( x \) increases.
### Visual Details:
The graph shows the intersection points with each segment of the function and illustrates the behavior of the function across the different defined intervals. Keep note of any discontinuities or breaks, which occur at \( x = -1 \) and \( x = 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f440a4a-a87c-43e8-9c30-e4890b6a6760%2F1e64155a-998b-4352-b5ce-b647d07a220b%2Fcs5ogbi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Function and Limits Analysis
### Given Piecewise Function:
Let
\[
g(x) =
\begin{cases}
-x & \text{if } x \leq -1 \\
1 - x^2 & \text{if } -1 < x < 1 \\
x - 1 & \text{if } x > 1
\end{cases}
\]
### (a) Evaluate the following limits, if they exist. (If an answer does not exist, enter DNE.)
1. (i) \( \lim_{x \to 1^+} g(x) \)
[Input Box]
2. (ii) \( \lim_{x \to 1} g(x) \)
[Input Box]
3. (iii) \( \lim_{x \to 0} g(x) \)
[Input Box]
4. (iv) \( \lim_{x \to -1} g(x) \)
[Input Box]
5. (v) \( \lim_{x \to -1^+} g(x) \)
[Input Box]
6. (vi) \( \lim_{x \to -1^-} g(x) \)
[Input Box]
### (b) Sketch the graph of \( g \).
### Graph Explanation:
The graph includes three distinct parts based on the piecewise function definition:
- **For \( x \leq -1 \)**: The section of the graph represents the linear function \( -x \). It is a straight line with a negative slope, leading downwards as \( x \) decreases.
- **For \( -1 < x < 1 \)**: This part displays the quadratic function \( 1 - x^2 \). The graph within this interval is a downward-opening parabola with its vertex at \( (0, 1) \).
- **For \( x > 1 \)**: Here, the graph corresponds to the linear function \( x - 1 \), which is a line with a positive slope, moving upwards as \( x \) increases.
### Visual Details:
The graph shows the intersection points with each segment of the function and illustrates the behavior of the function across the different defined intervals. Keep note of any discontinuities or breaks, which occur at \( x = -1 \) and \( x = 1
Transcribed Image Text:## Transcription of Graphs for Educational Website
### Problem Statement
(b) Sketch the graph of \( g \).
### Graph Descriptions
#### Graph 1 (Upper Left)
- **Axes**: The x-axis ranges from -3 to 3, and the g(x)-axis ranges from -3 to 3.
- **Curve**: There is a continuous curve, resembling a parabola turned sideways, with an open interval on the left and closed on the right at x = 1.
- **Intercepts and Points**:
- Closed circle at (1, 0).
- Open circle near (-1, 2) on the left segment.
#### Graph 2 (Upper Right)
- **Axes**: The x-axis ranges from -3 to 3, and the g(x)-axis ranges from -3 to 3.
- **Curve**: The graph dips and curves upward, with gaps.
- **Intercepts and Points**:
- Open circle at x = -1.
- Closed circle at (1, 1).
- An open circle is noted near the origin on the positive x-region.
#### Graph 3 (Lower Left)
- **Axes**: The x-axis ranges from -3 to 3, and the g(x)-axis ranges from -3 to 3.
- **Curve**: A curve is open at x = 1 and closed towards the negative x-values.
- **Intercepts and Points**:
- Closed circle at x = 1.
- Open circle at (-1, 0).
#### Graph 4 (Lower Right)
- **Axes**: The x-axis ranges from -3 to 3, and the g(x)-axis ranges from -3 to 3.
- **Curve**: Similar to the previous graphs, featuring discontinuities and a solid circle for closure.
- **Intercepts and Points**:
- Closed circle at (-1, 1).
- Open circle near x = 1 on the rising segment.
Each graph visually represents the function \( g(x) \) with variations in openness and linear slopes, showcasing behavior near specific points like x = -1 and x = 1.
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