Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.2 Limits and Continuity - 14.2 Exercises - Page 951: 40



Work Step by Step

Conversion of polar co-ordinates $(r, \theta)$ are: $x=r \cos \theta$ and $y= r \sin \theta$ Given: $\lim\limits_{(x,y) \to(0,0)}(x^2+y^2) \ln (x^2+y^2)$ This implies $=\lim\limits_{r \to0}(r^2 \cos^2 \theta+r^2 \sin^2 \theta) \ln (r^2 \cos^2 \theta+r^2 \sin^2 \theta)$ $=\lim\limits_{r \to 0}r^2 (\cos^2 \theta+\sin^2 \theta) \ln (r^2 \cos^2 \theta+r^2 \sin^2 \theta)$ $=\lim\limits_{r \to 0} r^2 \ln r^2$ $:=\lim\limits_{r \to 0}\dfrac{2/r}{-2/r^3}$ $=0$
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