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: 22



Work Step by Step

Consider $P(x,y)=xy$ Here, we have the point $P(x,y) \to O(0,0)$ This implies that $xy \to 0$ Let us consider $u=xy$ Then $\lim\limits_{u \to 0}\dfrac{1-\cos u}{u}=\dfrac{0}{0}$ This shows that the limit has Indeterminate form thus, we will have to apply L-Hospital's rule Thus, we get $\lim\limits_{u \to 0}\dfrac{\sin u}{(1)}=\sin (0)=0$
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