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: 65



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) } arctan (\dfrac{|x|+|y|}{x^2+x+y^2}$ or, $=\lim\limits_{r \to 0} arctan (\dfrac{ |r \cos \theta| +|r \sin \theta|}{r^2})$ or, $=\lim\limits_{r \to 0} arctan (\dfrac{ |\cos \theta| +|\sin \theta|}{r})$ or, $=arctan (\infty)$ or, $ =\dfrac{\pi}{2}$
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