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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![### Calculating the Limit of a Multivariable Function
**Question:**
Does the following limit exist? Justify your answer.
\[ \lim_{(x,y) \to (0,0)} \frac{2x^2 - xy^2}{x^2 + 2y^2} \]
**Explanation:**
1. First, we need to analyze the limit by approaching the point \((0,0)\) from different paths.
2. Let's begin by approaching along the x-axis (\(y = 0\)):
\[
\lim_{x \to 0} \frac{2x^2 - x \cdot 0^2}{x^2 + 2 \cdot 0^2} = \lim_{x \to 0} \frac{2x^2}{x^2} = \lim_{x \to 0} 2 = 2
\]
3. Next, approach along the y-axis (\(x = 0\)):
\[
\lim_{y \to 0} \frac{2 \cdot 0^2 - 0 \cdot y^2}{0^2 + 2y^2} = \lim_{y \to 0} \frac{0}{2y^2} = 0
\]
4. Since these two paths yield different limits (2 and 0), the limit does not exist.
**Conclusion:**
The limit does not exist because approaching \((0,0)\) along different paths results in different values. Thus, we cannot define a unique limit for the given function as \((x,y) \to (0,0)\).
#### Visual Representation
If necessary, you can visualize the problem by plotting the function \(\frac{2x^2 - xy^2}{x^2 + 2y^2}\) in a 3D graph. The graph will illustrate how the function behaves near the point \((0,0)\), further supporting the conclusion that the limit is path-dependent and does not exist.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedee5435-55e0-4ec8-aecd-101c845dd6aa%2F2c330a59-0377-4f17-9ac8-8ca4902efdae%2Fk6btew_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Limit of a Multivariable Function
**Question:**
Does the following limit exist? Justify your answer.
\[ \lim_{(x,y) \to (0,0)} \frac{2x^2 - xy^2}{x^2 + 2y^2} \]
**Explanation:**
1. First, we need to analyze the limit by approaching the point \((0,0)\) from different paths.
2. Let's begin by approaching along the x-axis (\(y = 0\)):
\[
\lim_{x \to 0} \frac{2x^2 - x \cdot 0^2}{x^2 + 2 \cdot 0^2} = \lim_{x \to 0} \frac{2x^2}{x^2} = \lim_{x \to 0} 2 = 2
\]
3. Next, approach along the y-axis (\(x = 0\)):
\[
\lim_{y \to 0} \frac{2 \cdot 0^2 - 0 \cdot y^2}{0^2 + 2y^2} = \lim_{y \to 0} \frac{0}{2y^2} = 0
\]
4. Since these two paths yield different limits (2 and 0), the limit does not exist.
**Conclusion:**
The limit does not exist because approaching \((0,0)\) along different paths results in different values. Thus, we cannot define a unique limit for the given function as \((x,y) \to (0,0)\).
#### Visual Representation
If necessary, you can visualize the problem by plotting the function \(\frac{2x^2 - xy^2}{x^2 + 2y^2}\) in a 3D graph. The graph will illustrate how the function behaves near the point \((0,0)\), further supporting the conclusion that the limit is path-dependent and does not exist.
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