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2 x³y² 3. We want to investigate lim (x,y)-(0,0) x² + y² We'll try approaching (0,0) along several different paths. Show your work on each one. a. What limit do you get when you approach the origin along the yaxis? b. What limit do you get when you approach the origin along an arbitrary straight line, y=mx? c. What limit do you get when you approach the origin along a parabola such as y=x²?

2 x³y² 3. We want to investigate lim (x,y)-(0,0) x² + y² We'll try approaching (0,0) along several different paths. Show your work on each one. a. What limit do you get when you approach the origin along the yaxis? b. What limit do you get when you approach the origin along an arbitrary straight line, y=mx? c. What limit do you get when you approach the origin along a parabola such as y=x²?

Calculus: Early Transcendentals
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 3 We want to investigate Lim…
**Part III: Calculating the Limits** 1. \(\lim_{(x,y) \to (-1, 1)} e^{-xy} \cos(x+y)\) 2. \(\lim_{(x,y) \to (1,0)} \ln \left( \frac{1 + y^2}{x^2 + xy} \right)\) 3. \(\lim_{(x,y) \to (0,0)} \frac{x^4 - y^4}{x^2 + y^2}\) **Level Curves and Limits** The level curves of \( z = S(x,y) \) are shown in the image below. Use the information from the level curves to determine the limits: a. \(\lim_{(x,y) \to (3,3)} S(x,y)\) b. \(\lim_{(x,y) \to (4,1)} S(x,y)\) c. \(\lim_{(x,y) \to (1,2)} S(x,y)\) d. \(\lim_{(x,y) \to (2,2)} S(x,y)\) *Explanation of Graph:* The graph shows several labeled level curves of the function \( z = S(x,y) \). Each curve represents a constant value of \( z \). The curves are labeled with values such as \( z = 1 \), \( z = 2 \), \( z = 3 \), and \( z = 4 \), and they provide information on how the function \( S(x,y) \) behaves in the \( x \) and \( y \) coordinate plane. **Exploring Limits Approaching (0, 0) Along Different Paths** We want to investigate \(\lim_{(x,y) \to (0,0)} \frac{x^3 y^2}{x^4 + y^8}\). Let's approach (0,0) along different paths and show the work: a. What limit do you get when you approach the origin along the \(\text{y-axis}\)? b. What limit do you get when you approach the origin along an arbitrary straight line, \( y = mx \)? c. What limit do you get when you approach the origin along a parabola such as \( y = x^2 \)? d. What is your prediction for the value of \(\lim_{(x
Transcribed Image Text:**Part III: Calculating the Limits** 1. \(\lim_{(x,y) \to (-1, 1)} e^{-xy} \cos(x+y)\) 2. \(\lim_{(x,y) \to (1,0)} \ln \left( \frac{1 + y^2}{x^2 + xy} \right)\) 3. \(\lim_{(x,y) \to (0,0)} \frac{x^4 - y^4}{x^2 + y^2}\) **Level Curves and Limits** The level curves of \( z = S(x,y) \) are shown in the image below. Use the information from the level curves to determine the limits: a. \(\lim_{(x,y) \to (3,3)} S(x,y)\) b. \(\lim_{(x,y) \to (4,1)} S(x,y)\) c. \(\lim_{(x,y) \to (1,2)} S(x,y)\) d. \(\lim_{(x,y) \to (2,2)} S(x,y)\) *Explanation of Graph:* The graph shows several labeled level curves of the function \( z = S(x,y) \). Each curve represents a constant value of \( z \). The curves are labeled with values such as \( z = 1 \), \( z = 2 \), \( z = 3 \), and \( z = 4 \), and they provide information on how the function \( S(x,y) \) behaves in the \( x \) and \( y \) coordinate plane. **Exploring Limits Approaching (0, 0) Along Different Paths** We want to investigate \(\lim_{(x,y) \to (0,0)} \frac{x^3 y^2}{x^4 + y^8}\). Let's approach (0,0) along different paths and show the work: a. What limit do you get when you approach the origin along the \(\text{y-axis}\)? b. What limit do you get when you approach the origin along an arbitrary straight line, \( y = mx \)? c. What limit do you get when you approach the origin along a parabola such as \( y = x^2 \)? d. What is your prediction for the value of \(\lim_{(x
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