Answer
The limit is
$$\lim_{(x,y)\to(a,b)}(f(x,y)g(x,y))=12.$$
Work Step by Step
It is given that
$$\lim_{(x,y)\to(a,b)}f(x,y)=4,\quad \lim_{(x,y)\to(a,b)}g(x,y)=3.$$
To find the required limit we will use that the limit of the product is a product of limits:
$$\lim_{(x,y)\to(a,b)}(f(x,y)g(x,y)) = \lim_{(x,y)\to(a,b)}f(x,y)\lim_{(x,y)\to(a,b)}g(x,y)=4\cdot3=12.$$