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Evaluate the following: We need to calculate the limit of the function f(x, y) as (x, y) approaches (0, 0), where f(x, y) is defined as f(x, y) xy² x² + y². = First Method: For (x, y) ‡ (0, 0), we observe that 0 ≤ ²¹+y² ≤ 1 because the numerator is not greater than the xy² denominator. Multiplying both sides by |x| yields 0 ≤ 2² ≤ |x|. By using the Squeeze Theorem, xy² = 0 since lim(x,y) →(0,0) |x| = 0. we conclude that lim(x,y)→(0,0) x²+y² Second Method:

Evaluate the following: We need to calculate the limit of the function f(x, y) as (x, y) approaches (0, 0), where f(x, y) is defined as f(x, y) xy² x² + y². = First Method: For (x, y) ‡ (0, 0), we observe that 0 ≤ ²¹+y² ≤ 1 because the numerator is not greater than the xy² denominator. Multiplying both sides by |x| yields 0 ≤ 2² ≤ |x|. By using the Squeeze Theorem, xy² = 0 since lim(x,y) →(0,0) |x| = 0. we conclude that lim(x,y)→(0,0) x²+y² Second Method:

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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Evaluate the following: We need to calculate the limit of the function f(x, y) as (x, y) approaches (0, 0), where f(x, y) is defined as f(x, y) = x² + y². xy² First Method: For (x, y) = (0,0), we observe that 0 ≤ 2+² ≤ 1 because the numerator is not greater than the xy² denominator. Multiplying both sides by a yields 0 ≤ ² ≤ x. By using the Squeeze Theorem, · 0 since lim(x,y)→(0,0) |x| = 0. we conclude that lim (z,y)→(0,0) 72²2 xy² = x²+y² Second Method: = Using polar coordinates, we express x and y as x = r cos 0 and y = r sin 0, respectively. Then x² + y² = r², and for r = 0, we have 0 ≤ xy² (r cos 0) (r sin 0)² r cos 0 sin² 0 ≤r. As (x, y) x² + y² approaches (0, 0), the variable r also approaches 0. Again, applying the Squeeze Theorem, we conclude that lim(x,y)→(0,0) x²+y² xy² = 0. = In both methods, we obtain the same result, namely that the limit of f(x, y) as (x, y) approaches (0, 0) is 0. The approach is correct The approach is incorrect The approach is almost acceptable
Transcribed Image Text:Evaluate the following: We need to calculate the limit of the function f(x, y) as (x, y) approaches (0, 0), where f(x, y) is defined as f(x, y) = x² + y². xy² First Method: For (x, y) = (0,0), we observe that 0 ≤ 2+² ≤ 1 because the numerator is not greater than the xy² denominator. Multiplying both sides by a yields 0 ≤ ² ≤ x. By using the Squeeze Theorem, · 0 since lim(x,y)→(0,0) |x| = 0. we conclude that lim (z,y)→(0,0) 72²2 xy² = x²+y² Second Method: = Using polar coordinates, we express x and y as x = r cos 0 and y = r sin 0, respectively. Then x² + y² = r², and for r = 0, we have 0 ≤ xy² (r cos 0) (r sin 0)² r cos 0 sin² 0 ≤r. As (x, y) x² + y² approaches (0, 0), the variable r also approaches 0. Again, applying the Squeeze Theorem, we conclude that lim(x,y)→(0,0) x²+y² xy² = 0. = In both methods, we obtain the same result, namely that the limit of f(x, y) as (x, y) approaches (0, 0) is 0. The approach is correct The approach is incorrect The approach is almost acceptable
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