Word Count : 291

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To check the continuity of ( f(x, y) ) at ((0,0)), consider the limit [ \lim_{(x,y) \to (0,0)} \frac{2x^3 y}{x^2 + y^2}. ] Using polar coordinates, set ( x = r \cos \theta ), ( y = r \sin \theta ). Then [ f(x, y) = \frac{2 (r \cos \theta)^3 (r \sin \theta)}{(r \cos \theta)^2 + (r \sin _______ ____ ____ ________ ________ ______ _______ ___ ___ ___.
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