If the dark energy density asymptotically approaches a nonzero constant , \rho _ { DE } \rightarrow \rho _ { 0 } , then its equation of state parameter w necessarily approaches -1 .
The converse is not true ; dark energy with w \rightarrow - 1 can correspond to either \rho _ { DE } \rightarrow \rho _ { 0 } or \rho _ { DE } \rightarrow 0 .
This provides a natural division of models with w \rightarrow - 1 into two distinct classes : asymptotic \Lambda ( \rho _ { DE } \rightarrow \rho _ { 0 } ) and pseudo- \Lambda ( \rho _ { DE } \rightarrow 0 ) .
We delineate the boundary between these two classes of models in terms of the behavior of w ( a ) , \rho _ { DE } ( a ) , and a ( t ) .
We examine barotropic and quintessence realizations of both types of models .
Barotropic models with positive squared sound speed and w \rightarrow - 1 are always asymptotically \Lambda ; they can never produce pseudo- \Lambda behavior .
Quintessence models can correspond to either asymptotic \Lambda or pseudo- \Lambda evolution , but the latter is impossible when the expansion is dominated by a background barotropic fluid .
We show that the distinction between asymptotic \Lambda and pseudo- \Lambda models for w > -1 is mathematically dual to the distinction between pseudo-rip and big/little rip models when w < -1 .