Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Section 14.2 - Limits and Continuity in Higher Dimensions - Exercises 14.2 - Page 798: 67


$\ln 3$

Work Step by Step

The polar-coordinates are defined as: $x= r \cos \theta , y = r \sin \theta$ and $r^2=x^2+y^2$ $ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \ln (\dfrac{3x^2-x^2y^2+3y^2}{x^2+y^2}$ or, $=\lim\limits_{r \to 0} \ln (\dfrac{ 3r^2 \cos^2 \theta- r^2 \cos^2 \theta r^2 \sin^2 \theta+3r^2 \sin^2 \theta}{r^2})$ or, $=\lim\limits_{r \to 0} \ln (3\cos^2 \theta-r^2 \cos ^2 \theta \times \sin ^2 \theta+3 \sin ^2 \theta)$ or, $=\lim\limits_{r \to 0} \ln [(3) (\cos^2 \theta + \sin ^2 \theta)]$ or, $=\ln 3$
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