Dark energy with $w \rightarrow -1$: Asymptotic $\Lambda$ versus
  pseudo-$\Lambda$
  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 $w < -1$.
