I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject).

For the kind of example I have in mind, consider the exponential function as a convergent power series e^x=1+x+x^2/2!+... on the interval [-1;1] say, and consider the varieties A0:={(x,y)\in\R^2 s.t. y=1}, A1:={(x,y)\in\R^2 s.t. y=1+x}, A2:={(x,y)\in\R^2 s.t. y=1+x+x^2/2!}... which converge (in the euclidean topology) to the graph of the exponential (as in the picture here).