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: 21

Answer

limit does not exist

Work Step by Step

Given: $\lim\limits_{(x,y,z) \to (0,0,0)}\frac{xy+yz^{2}+xz^{2}}{ {x^{2}+y^{2}+z^{4}}}$ 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 $y=0,z=0$ $=\lim\limits_{(x,y,z) \to (0,0,0)}\frac{0+0+0}{ {x^{2}+0^{2}+0^{4}}}=\frac{0}{x^{2}}=0$ Approach the point (0, 0, 0) along the curve where $x=y$ and $x=z^{2}$ $\lim\limits_{(x,y,z) \to (0,0,0)}\frac{x^{2}+x^{2}+x^{2}}{ {x^{2}+x^{2}+x^{2}}}=\frac{0}{x^{2}}$ $=\lim\limits_{(x,y,z) \to (0,0,0)}\frac{3x^{2}}{3x^{2}}$ $=1$ For a limit to exist, all the paths must converge to the same point.Since, function $f(x,y,z)$ has two different values along two different paths, it follows that limit does not exist, Hence, the limit does not exist.
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