Word Count : 635

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We are tasked with checking the continuity and differentiability of the function: \[ f(x, y) = \begin{cases} \frac{2x^3y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases} \] ### Step 1: Checking Continuity at (0, 0) A function \( f(x, y) \) is continuous at a point \((a, b)\) if: \[ \lim_{(x, y) \to (a, b)} f(x, y) = f(a, b) \] For \((0, 0)\), this means checking if: \[ \lim_{(x, y) \to (0, 0)} f(x, y) = f(0, 0) \] Since \( f(0, 0) = 0 \), we need to verify if the limit exists and equals 0. We'll evaluate the limit along different paths to check if it is consistent. 1. Along the ___ ______ __________ ______ _________ __________ ____ _________ ________ _________ ___.
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