Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 34


Does not exist

Work Step by Step

Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{|x|}{|x|+|y|}$$ Consider the line $y=mx$ that passes through $(0,0)$: \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{|x|}{|x|+|y|}&=\lim _{(x, y) \rightarrow(0,0)} \frac{|x|}{|x|+|mx|}\\ &= \lim _{(x, y) \rightarrow(0,0)} \frac{1}{1+|m|}\\ \end{align*} Since the limit depends on $m$, then $\lim _{(x, y) \rightarrow(0,0)} \dfrac{|x|}{|x|+|y|}$ does not exist.
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.