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: 22

Answer

$0$

Work Step by Step

Given: $\lim\limits_{(x,y,z) \to (0,0,0)}\frac{x^{2}y^{2}z^{2}}{ {x^{2}+y^{2}+z^{2}}}$ We can approach the points (0, 0, 0) in space through the co-ordinate axes or through co-ordinate plane or through the symmetrical or unsymmetrical lines. Now, approach the point (0, 0, 0) along x-axis. To evaluate limit along x-axis; put $y=0,z=0$ $f(x,0,0)=\frac{x^{2}0^{2}0^{2}}{ {x^{2}+0^{2}+0^{2}}}=0$ To evaluate limit along y-axis; put $x=0,z=0$ $f(0,y,0)=\frac{0^{2}y^{2}0^{2}}{ {0^{2}+y^{2}+0^{2}}}=0$ To evaluate limit along z-axis; put $x=0,y=0$ $f(0,0,z)=\frac{0^{2}0^{2}z^{2}}{ {0^{2}+0^{2}+z^{2}}}=0$ Hence, the limit converges to $0$.
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