Answer
The limit is
$$\lim_{(x,y)\to(1,2)}e^{xy}=e^2.$$
The function is continuous on its' domain.
Work Step by Step
This limit is calculated by a simple substitution
$$\lim_{(x,y)\to(1,2)}e^{xy}=e^{1\cdot2}=e^2.$$
This function is continuous on its domain because for every ordered pair $(x_0,y_0)$ we have
$$\lim_{(x,y)\to(x_0,y_0)}e^{xy}=e^{x_0y_0}.$$
