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
Does the limit exist? Why or why not
![The given mathematical expression represents a two-dimensional limit of a function as the point \((x, y)\) approaches the origin \((0,0)\).
**Expression:**
\[ \lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2} \]
**Explanation:**
The limit provided is a function of two variables, \(x\) and \(y\). The expression \(\frac{xy}{x^2 + y^2}\) is evaluated as both \(x\) and \(y\) approach zero simultaneously.
### Key Components:
- **lim** indicates the limit of the function.
- \((x, y) \to (0, 0)\) specifies that the variables \(x\) and \(y\) are approaching the point \((0, 0)\).
- \(\frac{xy}{x^2 + y^2}\) is the function whose limit is being evaluated.
### Steps to Analyze the Limit:
1. Substitute \(x\) and \(y\) with values approaching zero along different paths to see if the limit is path-independent.
2. If the limit depends on the path taken to approach \((0, 0)\), the limit does not exist. If all paths yield the same result, the limit exists and is that common value.
### Potential Paths:
1. **Path \(y = 0\):**
- Substitution results in: \(\frac{x \cdot 0}{x^2 + 0^2} = 0\)
2. **Path \(x = 0\):**
- Substitution results in: \(\frac{0 \cdot y}{0^2 + y^2} = 0\)
3. **Path \(y = x\):**
- Substitution results in: \(\frac{x \cdot x}{x^2 + x^2} = \frac{x^2}{2x^2} = \frac{1}{2}\)
### Conclusion:
Since the limit along the paths \(y = 0\) and \(x = 0\) evaluates to \(0\) and the limit along the path \(y = x\) evaluates to \(\frac{1}{2}\), the limit depends on the path taken to approach \((0, 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd3efa0e7-ba10-45c1-a0b8-fb98ed40206a%2Fbf1952e9-5a2a-43ab-93d1-828509a71679%2Fcc3auj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The given mathematical expression represents a two-dimensional limit of a function as the point \((x, y)\) approaches the origin \((0,0)\).
**Expression:**
\[ \lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2} \]
**Explanation:**
The limit provided is a function of two variables, \(x\) and \(y\). The expression \(\frac{xy}{x^2 + y^2}\) is evaluated as both \(x\) and \(y\) approach zero simultaneously.
### Key Components:
- **lim** indicates the limit of the function.
- \((x, y) \to (0, 0)\) specifies that the variables \(x\) and \(y\) are approaching the point \((0, 0)\).
- \(\frac{xy}{x^2 + y^2}\) is the function whose limit is being evaluated.
### Steps to Analyze the Limit:
1. Substitute \(x\) and \(y\) with values approaching zero along different paths to see if the limit is path-independent.
2. If the limit depends on the path taken to approach \((0, 0)\), the limit does not exist. If all paths yield the same result, the limit exists and is that common value.
### Potential Paths:
1. **Path \(y = 0\):**
- Substitution results in: \(\frac{x \cdot 0}{x^2 + 0^2} = 0\)
2. **Path \(x = 0\):**
- Substitution results in: \(\frac{0 \cdot y}{0^2 + y^2} = 0\)
3. **Path \(y = x\):**
- Substitution results in: \(\frac{x \cdot x}{x^2 + x^2} = \frac{x^2}{2x^2} = \frac{1}{2}\)
### Conclusion:
Since the limit along the paths \(y = 0\) and \(x = 0\) evaluates to \(0\) and the limit along the path \(y = x\) evaluates to \(\frac{1}{2}\), the limit depends on the path taken to approach \((0, 0
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