Thomas' Calculus 13th Edition

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

Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 18



Work Step by Step

Consider two difference approaches to (0,0): 1. along y=-x, we have $\lim\limits_{x,y \to 0,0}\frac{sin(x-y)}{|x|+|y|}=\lim\limits_{x,y \to 0,0}\frac{sin(2x)}{2|x|}=1$ for $x\gt0$ 2. along y=2x, we have $\lim\limits_{x,y \to 0,0}\frac{sin(x-y)}{|x|+|y|}=\lim\limits_{x,y \to 0,0}\frac{sin(-x)}{3|x|}=-\frac{1}{3}$ for $x\gt0$ Hence, function $f(x,y)$ is not continuous at the origin.
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