Let $f(x, y) = \left(\dfrac{y\sin(x)}{x}, \tan\left(\dfrac{x}{y}\right), x^2+y^2-xy\right), \quad (a, b) = (0, 1)$. The question originally says that find the maximal domain and find the limits. But I wonder whether I can prove the limits exist with $\epsilon-\delta$ separately. I found that:
$$f(x,y) \rightarrow (0,0,1)$$
Let, $f_{1}(x,y)= \dfrac{y\sin(x)}{x}, \quad f_{2}(x, y) = \tan\left(\dfrac{x}{y}\right), \quad f_{3}(x, y) = x^2+y^2-xy$. Then, we can prove that the limits are exist since:
$$f_{1}(x,y) \rightarrow 0, \quad f_{2}(x,y) \rightarrow 0, \quad f_{3}(x,y) \rightarrow 1$$
So my question is, if we prove that those limits are exist (by $\epsilon- \delta$) then is it equal to proving $f(x,y) \rightarrow (0,0,1)$ by $\epsilon-\delta$? Thanks in advance!
If we have $y=(y_1,\cdots,y_n)=f(x)=(f^1(x),\cdots, f^n(x))$ with limit $A=(A_1,\cdots,A_n)$, then iequalities $$|y_i-A_i|\leqslant d(y,A) \leqslant \sqrt{n}\max\limits_{1\leqslant i \leqslant n}|y_i-A_i|$$ gives $$\lim f(x)=A \Leftrightarrow \lim f^i(x)=A_i$$