Find Two Paths of Approach from Which One Can Conclude $f ( x , y , z )$

question 89

Essay

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-We say that a function $f ( x , y , z )$ approaches the limit $L$ as $( x , y , z )$ approaches $( x 0 , y 0 , z 0 )$ and write
$( x , y , z ) \rightarrow \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right) \quad f ( x , y , z ) = L$
if for every number $\varepsilon > 0$ , there exists a corresponding number $\delta > 0$ such that for all $( x , y , z )$ in the domain of $f$ , $0 < \sqrt { \left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } + \left( z - z _ { 0 } \right) ^ { 2 } } < \delta \Rightarrow | f ( x , y , z ) - L | < \varepsilon$ . Show that the $\delta - \varepsilon$ requirement in this definition is equivalent to $0 < \left| x - x _ { 0 } \right| < \delta , 0 < \left| y - y _ { 0 } \right| < \delta$ , and $0 < \left| \mathrm { z } - \mathrm { z } _ { 0 } \right| < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) - \mathrm { L } | < \varepsilon$ .

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