  We study the compatibility of the action of the DAHA of type GL with two
inverse systems of polynomial rings obtained from the standard Laurent
polynomial representations. In both cases, the crucial analysis is that of the
compatibility of the action of the Cherednik operators. Each case leads to a
representation of a limit structure (the +/- stable limit DAHA) on a space of
almost symmetric polynomials in infinitely many variables (the standard
representation). As an application, we show that the defining representation of
the double Dyck path algebra arises from the standard representation of the
+stable limit DAHA.
