If a sequence converges to 0 in $\mathbb{R}^2$ and a function is 0 at every point of the sequence and at 0, is 0 an eigenvalue of the Jacobian?

Question

The function is from $\mathbb{R}^2$ to $\mathbb{R}^2$.

I have thought about possibly picking a subsequence which converges to 0 from a single direction, then taking the directional derivative, but I don't think this subsequence exists.