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: 16


limit $l$ does not exist

Work Step by Step

Consider $l=\lim\limits_{(x,y) \to (0,0)} \dfrac{x^2+y^2}{xy}$ Suppose $y=p x$; $p \ne 0$ Now, $l=\lim\limits_{(x,px) \to (0,0)} \dfrac{x^2+p^2 x^2}{px^2}$ Hence, $l=\lim\limits_{(x,px) \to (0,0)} \dfrac{1+p^2}{p}=\dfrac{1+p^2}{p}$ Hence, the limit $l$ does not exist as we can see that $l$ will have different limits for different values of $p$.
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