Compute $\lim_{(x,y) \to (0,0)} \frac{x^2 - 2\cos(y) + 2}{y^2 - 2\cos(x) + 2}$

Question

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.

Answer 1

Note that, since $\cos(t)=1-\frac{t^2}{2}+O(t^4)$ as $t\to 0$, we have that $$\frac{x^2 - 2\cos(y) + 2}{y^2 - 2\cos(x) + 2}=\frac{x^2 - 2+y^2+O(y^4) + 2}{y^2 - 2+x^2+O(x^4) + 2}=\frac{r^2+O(y^4)}{r^2+O(x^4)}=\frac{1+\frac{O(y^4)}{r^2}}{1+\frac{O(x^4)}{r^2}}$$ where $r^2=x^2+y^2$. Can you take it from here?