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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![## Graph Sketching Exercise
**Task:**
Sketch the graph of a function that satisfies ALL the following properties:
1. \[
\lim_{x \to \infty} f(x) = -2
\]
2. \[
\lim_{x \to -\infty} f(x) \text{ does not exist}
\]
3. \[
\lim_{x \to 2^{-}} f(x) = \infty
\]
4. \[
\lim_{x \to 2^{+}} f(x) = -\infty
\]
5. \[
f(2) = 0
\]
6. The function has exactly one vertical asymptote. Be creative!
### Explanation of Graph Properties
- The first condition (\(\lim_{x \to \infty} f(x) = -2\)) tells us that as \(x\) approaches positive infinity, the function \(f(x)\) approaches \(-2\).
- The second condition (\(\lim_{x \to -\infty} f(x) \text{ does not exist}\)) means that as \(x\) approaches negative infinity, the function \(f(x)\) has no specific limit; it could oscillate or diverge.
- The third condition (\(\lim_{x \to 2^{-}} f(x) = \infty\)) indicates that as \(x\) approaches \(2\) from the left-hand side, the function \(f(x)\) approaches positive infinity.
- The fourth condition (\(\lim_{x \to 2^{+}} f(x) = -\infty\)) denotes that as \(x\) approaches \(2\) from the right-hand side, the function \(f(x)\) approaches negative infinity.
- The fifth condition (\(f(2) = 0\)) implies that the function passes through the point (2, 0).
- The sixth condition requires the function to have exactly one vertical asymptote. This vertical asymptote is likely at \(x = 2\) considering the previous conditions.
Combining all these properties will guide you to sketch a function that meets the given requirements. Be creative while ensuring the function adheres to all specified limits and conditions!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4b64424-2f1e-400b-a2f3-ccc24d7d74e8%2Fb1f31cc2-72e3-42ad-bb39-b59fc42076cf%2Ftebm018_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Graph Sketching Exercise
**Task:**
Sketch the graph of a function that satisfies ALL the following properties:
1. \[
\lim_{x \to \infty} f(x) = -2
\]
2. \[
\lim_{x \to -\infty} f(x) \text{ does not exist}
\]
3. \[
\lim_{x \to 2^{-}} f(x) = \infty
\]
4. \[
\lim_{x \to 2^{+}} f(x) = -\infty
\]
5. \[
f(2) = 0
\]
6. The function has exactly one vertical asymptote. Be creative!
### Explanation of Graph Properties
- The first condition (\(\lim_{x \to \infty} f(x) = -2\)) tells us that as \(x\) approaches positive infinity, the function \(f(x)\) approaches \(-2\).
- The second condition (\(\lim_{x \to -\infty} f(x) \text{ does not exist}\)) means that as \(x\) approaches negative infinity, the function \(f(x)\) has no specific limit; it could oscillate or diverge.
- The third condition (\(\lim_{x \to 2^{-}} f(x) = \infty\)) indicates that as \(x\) approaches \(2\) from the left-hand side, the function \(f(x)\) approaches positive infinity.
- The fourth condition (\(\lim_{x \to 2^{+}} f(x) = -\infty\)) denotes that as \(x\) approaches \(2\) from the right-hand side, the function \(f(x)\) approaches negative infinity.
- The fifth condition (\(f(2) = 0\)) implies that the function passes through the point (2, 0).
- The sixth condition requires the function to have exactly one vertical asymptote. This vertical asymptote is likely at \(x = 2\) considering the previous conditions.
Combining all these properties will guide you to sketch a function that meets the given requirements. Be creative while ensuring the function adheres to all specified limits and conditions!
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