Find Two Paths of Approach from Which One Can Conclude $3 - x ^ { 2 } y ^ { 3 } < \frac { 3 \tan ^ { - 1 } x y } { x y } < 3$

question 339

Essay

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-Does knowing that $3 - x ^ { 2 } y ^ { 3 } < \frac { 3 \tan ^ { - 1 } x y } { x y } < 3$ tell you anything about $( x , y ) \rightarrow ( 0,0 ) \frac { 3 \tan ^ { - 1 } x y } { x y }$ ? Give reasons for your answer.

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Definitions:

Abbreviation

The shortened form of a word or phrase used to save space or avoid repetition.

LOM

Limitation of Motion, a term used to describe the restricted movement of a limb or joint.

Analgesic

A category of drugs used to achieve pain relief; also known as painkillers.

Pain

A negative sensory and emotional reaction related to either existing or impending bodily injury, or explained as if such injury has occurred.