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

Answer

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.
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