Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Section 14.2 - Limits and Continuity in Higher Dimensions - Exercises 14.2 - Page 796: 24



Work Step by Step

Re-arrange the given equation as follows: $\lim\limits_{(x,y) \to (2,2)}\dfrac{(x-y)}{(x^4-y^4)}=\lim\limits_{(x,y) \to (2,2)}\dfrac{(x-y)}{(x^2)^2-(y^2)^2}$ This implies that $\lim\limits_{(x,y) \to (2,2)}\dfrac{(x-y)}{(x^2)^2-(y^2)^2}=\lim\limits_{(x,y) \to (2,2)}\dfrac{(x-y)}{(x-y)(x+y)(x^2+y^2)}=\lim\limits_{(x,y) \to (2,2)}\dfrac{1}{(x+y)(x^2+y^2)}$ Thus, $\lim\limits_{(x,y) \to (2,2)}\dfrac{1}{(x+y)(x^2+y^2)}=\dfrac{1}{(4)(8)}=\dfrac{1}{32}$
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