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
![**Finding the Limit:**
In this exercise, you are asked to find the limit of the given function as the point \((x, y)\) approaches \((3, 3)\).
\[
\lim_{(x, y) \to (3, 3)} \frac{2x^4 - 2y^4}{x^2 - y^2}
\]
**Steps to Approach the Problem:**
1. **Substitution:** Initially attempt to substitute \(x = 3\) and \(y = 3\) directly into the function. If you encounter a zero denominator, further analytical steps are necessary.
2. **Factoring:** Factor both the numerator and denominator when possible to find and eliminate common factors.
3. **Simplify:** Simplify the resulting expression to see if the limit can be evaluated more clearly.
**Graph or Diagram Explanation:**
If there were graphs or diagrams associated with this limit problem, they would typically illustrate:
- The behavior of the function \(f(x, y)\) near the point \((3, 3)\).
- How the function values converge to the limit from different paths, for instance, approaching along the line \(y=x\) or along \(y=3\).
**Example Path Analysis:**
- Substituting \(y = 3\):
\[
f(x, 3) = \frac{2x^4 - 2 \cdot 3^4}{x^2 - 3^2} = \frac{2x^4 - 162}{x^2 - 9}
\]
- Further simplification and analysis would follow from here.
This interactive exercise demands critical algebraic manipulation to reach a conclusive answer. Once you have worked through the problem, please submit your final answer in the provided answer box.
**Submit Question:**
There is a button labeled "Submit Question" which should be clicked once you have entered your answer in the box provided.
**Formula Context:**
Understanding the concepts of limits, algebraic manipulation, and L'Hopital's Rule will be beneficial when solving limit problems such as this. Make sure to practice different methods to gain proficiency.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2bc9f06-7337-4a04-9aea-9ce8610fc600%2Faccbc009-ce16-4c87-8d22-55a09b255159%2Fzqk1lvj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Limit:**
In this exercise, you are asked to find the limit of the given function as the point \((x, y)\) approaches \((3, 3)\).
\[
\lim_{(x, y) \to (3, 3)} \frac{2x^4 - 2y^4}{x^2 - y^2}
\]
**Steps to Approach the Problem:**
1. **Substitution:** Initially attempt to substitute \(x = 3\) and \(y = 3\) directly into the function. If you encounter a zero denominator, further analytical steps are necessary.
2. **Factoring:** Factor both the numerator and denominator when possible to find and eliminate common factors.
3. **Simplify:** Simplify the resulting expression to see if the limit can be evaluated more clearly.
**Graph or Diagram Explanation:**
If there were graphs or diagrams associated with this limit problem, they would typically illustrate:
- The behavior of the function \(f(x, y)\) near the point \((3, 3)\).
- How the function values converge to the limit from different paths, for instance, approaching along the line \(y=x\) or along \(y=3\).
**Example Path Analysis:**
- Substituting \(y = 3\):
\[
f(x, 3) = \frac{2x^4 - 2 \cdot 3^4}{x^2 - 3^2} = \frac{2x^4 - 162}{x^2 - 9}
\]
- Further simplification and analysis would follow from here.
This interactive exercise demands critical algebraic manipulation to reach a conclusive answer. Once you have worked through the problem, please submit your final answer in the provided answer box.
**Submit Question:**
There is a button labeled "Submit Question" which should be clicked once you have entered your answer in the box provided.
**Formula Context:**
Understanding the concepts of limits, algebraic manipulation, and L'Hopital's Rule will be beneficial when solving limit problems such as this. Make sure to practice different methods to gain proficiency.
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