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 950: 10


limits does not exist.

Work Step by Step

We notice that if we directly substitute limits in the given function $f(x,y)=\frac{5y^{4}cos^{2}x}{x^{4}+y^{4}}$ Then $f(0,0)=\frac{0}{0}$ We get intermediate form. Therefore, we will calculate limit of function in following way. To evaluate limit along x-axis; put $y=0$ $f(x,0)=\frac{5y^{4}cos^{2}x}{x^{4}+y^{4}}=\frac{5(0)^{4}cos^{2}x}{x^{4}+0}=0$ To evaluate limit along y-axis; put $x=0$ $f(0,y)=\frac{5y^{4}cos^{2}x}{x^{4}+y^{4}}=5$ For a limit to exist, all the paths must converge to the same point. Hence, both the limits are different and follow different paths, thus, given limits does not exist.
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