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: 15


Limit $l$ does not exist.

Work Step by Step

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

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.