Does the limit $$\lim_{(x,y) \to (0,0)} \left( \frac{x^2 - 2\cos(y) + 2}{y^2 - 2\cos(x) + 2} \right) $$ exist?
I think it does and it's equal to $1$, but I don't know how to prove it. I tried to use Taylor expansion of $\cos(x)$ and $\cos(y)$, but it doesn't help me to compute the limit.