Any functions $f(x, y)$ so that two limits exist

Question

I need to find for any functions $f(x, y)$ so that $$\lim_{\{x, y\}\rightarrow\{0, 0\}}\frac{f(x, y)- \frac{y}{x}}{1+ \frac{f(x, y)y}{x}}, \lim_{\{x, y\}\rightarrow\{0, 0\}}f(x, y){\,\,\it{exist}}$$ I used Wolfram|Alpha to find, so many choices, I spent my times, but maybe there's no $f(x, y)$ at all

I use that for the following calculating $$\lim_{\{x, y\}\rightarrow\{0, 0\}}x\tan^{-1}\frac{y}{x}$$

Answer 1

Let $f(0,0):=\lim f(x,y)$.

If we look at the first limit along the path $y=0$, it evaluates to $f(0,0)$. If we look at the first limit along the path $y=x$, it evaluates to $\frac{f(0,0)-1}{1+f(0,0)}$. If we look at the first limit along the path $y=-x$, it evaluates to $\frac{f(0,0)+1}{1-f(0,0)}$. If all these shall be equal, we must have $f(0,0)=\pm i$.