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: 22

Answer

Does not exist

Work Step by Step

Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{4}+x^{2} y^{2}+y^{4}}$$ Consider the line $y=mx$ that passes through $(0,0)$: \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{4}+x^{2} y^{2}+y^{4}}&=\lim _{x \rightarrow0} \frac{x^{4}-m^{4}x^4}{x^{4}+x^{2} m^{2}x^2+m^{4}x^4}\\ &=\lim _{x \rightarrow0} \frac{1-m^{4}}{1+ m^{2}+m^{4}}\\ &=\frac{1-m^{4}}{1+ m^{2}+m^{4}} \end{align*} Since the limit depends on $m$, then $\lim _{(x, y) \rightarrow(0,0)} \dfrac{x^{4}-y^{4}}{x^{4}+x^{2} y^{2}+y^{4}}$ does not exist.
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