Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 16


See the proof below.

Work Step by Step

Along the lines $ y=mx $, we have $$ f(x,y)=\frac{2x^2+3y^2}{x y }=\frac{2x^2+3m^2x^2}{m x^2}=\frac{2+3m^2}{m^2}.$$ Hence, along the paths $ y=mx $, we have $$ \lim\limits_{(x,y) \to (0,0)}\frac{2x^2+3y^2}{x y } =\frac{2+3m^2}{m^2}.$$ hence the limit depends on the slope $ m $ and so it does not exist.
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