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Suppose the functions $f:\mathbb R^2 \rightarrow \mathbb R$ and $g:\mathbb R^2 \rightarrow \mathbb R$ are continuously differentiable. Find necessary and sufficient conditions for these functions to be first-order approximations of each other at point (0,0).

My attempt: Using the first-order approximation at point (0,0) we have: $$f(\vec x+\vec h)=f(\vec x)+<\nabla f(\vec x),\vec h>=f(0,0)+f_x(0,0)x+f_y(0,0)y$$ $$g(x,y)=g(0,0)+g_x(0,0)x+g_y(0,0)y.$$

I want to connect these approximations, since we're asked to find necessary and sufficient conditions for these functions to be first-order approximations of each other at point (0,0), but how to connect these 2 equations? Or maybe my approach is not correct?

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