Directional derivative only for certain directionvectors

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How should i do this, i'm lost? Pointers but also full solutions are welcome.

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Hint

Change the Cartesian coordinates to polar coordinates as $$x_1=r\cos\phi\\x_2=r\sin\phi$$and rewrite the limit as a function of $\phi$ when $r\to 0$. For what values of $\phi$ does the limit exist?