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: 62



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) } \cos \dfrac{x^3-xy^2}{x^2+y^2} \\=\lim\limits_{r \to 0} \cos (\dfrac{ r^3 \cos^3 \theta- r^3 \sin^3 \theta}{r^2}) \\\lim\limits_{r \to 0} \cos [r^3 (\dfrac{ \cos^3 \theta- \sin^3 \theta}{r^2}] \\=\cos 0 \\=1$
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