Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $lim_{x\to x_0}f(x,y)$ does not exist

Question

Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $$lim_{y\to y_0}f(x,y)$$ and $$lim_{x\to x_0}(lim_{y\rightarrow y_0}f(x,y))$$ exists but $lim_{x\to x_0}f(x,y)$ does not exist for a fixed $(x_0,y_0)\in D$

I haven´t been able to give an example. Can you please help me? I would really appreciate it :)

Answer 1

$$f(x,y)=y\sin\left(\frac1x\right)$$ at $(0,0)$. (You can define $f(0,0)=0$, for example).

Answer 2

At $(x_0,y_0)=(0,0)$,$$(x,y)\mapsto \frac{xy}{x^2 + y^2}\ ?$$

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Edit: Following Matthias's comment -- this does not answer the question. I was thinking of a non-existing limit when $(x,y)\to (x_0,y_0)$.