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
![### Limit Analysis via Multiple Paths
**Problem Statement:**
4. Show that the following limit does not exist by using two paths \( y = 0 \) and \( x = -y \).
\[
\lim_{(x,y) \to (0,0)} \left( \frac{x^2}{x^2 + y^2} \right)
\]
**Explanation:**
To demonstrate that the limit does not exist, we must evaluate the expression along different paths approaching the origin \((0,0)\) in the coordinate plane and show that they yield different results. If the limits along these paths differ, the overall limit does not exist.
1. **Path 1: \( y = 0 \)**
Substitute \( y = 0 \) into the expression:
\[
\frac{x^2}{x^2 + 0^2} = \frac{x^2}{x^2} = 1
\]
As \((x,y)\) approaches \((0,0)\) on the line \( y = 0 \), the limit is 1.
2. **Path 2: \( x = -y \)**
Substitute \( x = -y \) into the expression:
\[
\frac{(-y)^2}{(-y)^2 + y^2} = \frac{y^2}{y^2 + y^2} = \frac{y^2}{2y^2} = \frac{1}{2}
\]
As \((x,y)\) approaches \((0,0)\) on the line \( x = -y \), the limit is \(\frac{1}{2}\).
Since the limits along the paths \( y = 0 \) and \( x = -y \) are 1 and \(\frac{1}{2}\) respectively, the limit of the function as \((x,y)\) approaches \((0,0)\) does not exist. Different paths yield different results, confirming that there is no unique limit at this point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf5aa2e5-f0b0-4839-a4ef-6f98d6dc8551%2F6683a10e-f8b2-4b9a-a96a-7498ab4c12e4%2Fog0ti1j_processed.png&w=3840&q=75)
Transcribed Image Text:### Limit Analysis via Multiple Paths
**Problem Statement:**
4. Show that the following limit does not exist by using two paths \( y = 0 \) and \( x = -y \).
\[
\lim_{(x,y) \to (0,0)} \left( \frac{x^2}{x^2 + y^2} \right)
\]
**Explanation:**
To demonstrate that the limit does not exist, we must evaluate the expression along different paths approaching the origin \((0,0)\) in the coordinate plane and show that they yield different results. If the limits along these paths differ, the overall limit does not exist.
1. **Path 1: \( y = 0 \)**
Substitute \( y = 0 \) into the expression:
\[
\frac{x^2}{x^2 + 0^2} = \frac{x^2}{x^2} = 1
\]
As \((x,y)\) approaches \((0,0)\) on the line \( y = 0 \), the limit is 1.
2. **Path 2: \( x = -y \)**
Substitute \( x = -y \) into the expression:
\[
\frac{(-y)^2}{(-y)^2 + y^2} = \frac{y^2}{y^2 + y^2} = \frac{y^2}{2y^2} = \frac{1}{2}
\]
As \((x,y)\) approaches \((0,0)\) on the line \( x = -y \), the limit is \(\frac{1}{2}\).
Since the limits along the paths \( y = 0 \) and \( x = -y \) are 1 and \(\frac{1}{2}\) respectively, the limit of the function as \((x,y)\) approaches \((0,0)\) does not exist. Different paths yield different results, confirming that there is no unique limit at this point.
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