University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 13 - Section 13.2 - Limits and Continuity in Higher Dimensions - Exercises - Page 693: 61

Answer

$0$

Work Step by Step

Use polar-coordinates: $x= r \cos \theta , y = r \sin \theta \\ r^2=x^2+y^2$ Now, $$ \lim\limits_{(x,y) \to (0,0) } |f(x,y)|=\lim\limits_{(x,y) \to (0,0) } \dfrac{x^3-xy^2}{x^2+y^2}\\ \\=\lim\limits_{r \to 0} \dfrac{ r^3 \cos^3 \theta- r \cos \theta \times r^2 \cos^2 \theta}{r^2} \\=\lim\limits_{r \to 0} r \cos \theta \times ( \cos^2 \theta - \sin^2 \theta ) \\=0$$
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