To evaluate is most likely easiest solved by converting to polars, because by converting to polars, it doesn't matter what path you take, you will always be making , so it transforms the limit into a limit of one variable instead of two. Anyway, with and , we find

Edit: I hope this post has been helpful, though I now realise I should have read the question in full, because when skimming through I thought you were asking to show the continuity at the point (x, y) = (0, 0), instead of what I now realise you were actually asking, i.e. to evaluate the partial derivatives.