University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 13 - Section 13.2 - Limits and Continuity in Higher Dimensions - Exercises - Page 691: 41


Limit does not exist.

Work Step by Step

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