OP, do you mean $\displaystyle \lim_{(x,y) \rightarrow (0, 0)} \sin(x)/x + y$ or $\displaystyle \lim_{(x,y) \rightarrow (0, 0)} sin(x)/(x + y)$?

The question is not about (0, 0) (we all know that the function value at the limit point is irrelevant to the definition of the limit), but about points (x, y) where x = -y (if the function is indeed sin(x) / (x + y)).

It's reasonable to assume that the domain consists of all points where the ratio sin(x) / x (or sin(x) / (x + y)) is defined.

According to

Wikipedia and

Encyclopedia of Mathematics, the answer is no. The definition of limit takes the intersection of a neighborhood of the limit point and the domain of the function.

That said, $\displaystyle \lim_{(x,y) \rightarrow (0, 0)} sin(x)/(x + y)$ does not exist. Indeed, the limit is 1 when (x, y) approaches (0, 0) along the x-axis. On the other hand, every open circle around (0, 0) has points (x, y) where |sin(x) / (x + y)| is arbitrarily large (fix some x and make y approach -x).