Difficult# Limits: Behavior Of $x^2y/(x^4+y^2)$ Near The Origin

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For every point $(x,y)\ne (0,0)$ in the punctured Cartesian plane, let:

$$ f(x,y) = \cfrac{x^2y}{x^4+y^2}.$$

Select each of the correct statements (and only the correct statements) about the function $f$ from the options below.