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: 41


limit does not exist

Work Step by Step

Here, we have $f(x,y)=-\dfrac{x}{\sqrt{x^2+y^2}}$ Let us consider one approach : $(x,y) \to (0,0)$ along $y=kx$ Then, we get $\lim\limits_{(x,y) \to (0,0)}-\dfrac{x}{\sqrt{x^2+(kx)^2}}=\lim\limits_{(x,y) \to (0,0)}-\dfrac{1}{\sqrt{1+k^2}}$ This shows that there are multiple limit values, therefore, and so, the limit does not exist at the point (0,0) for the function $f(x,y)=-\dfrac{x}{\sqrt{x^2+y^2}}$.
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