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:

Respiratory Membrane

The respiratory membrane is the thin barrier in the lungs where gas exchange occurs, allowing oxygen to enter the blood and carbon dioxide to be expelled.

Diffusion Coefficient

Measure of how easily a gas diffuses through a liquid or tissue.

Surface Tension

The elastic tendency of a fluid surface which makes it acquire the least surface area possible.

Lung Recoil

Decrease in the size of an expanded lung as a result of a decrease in the size (volume) of its alveoli; due to elastic recoil of elastic fibers surrounding alveoli and water surface tension of a thin film of water within alveoli.