Sketch the graph of a function that satisfies ALL the following properties:

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!
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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