  We derive slow-roll conditions for thawing k-essence with a separable
Lagrangian $p(X,\phi)=F(X)V(\phi)$. We examine the evolution of the equation of
state parameter, $w$, as a function of the scale factor $a$, for the case where
$w$ is close to -1. We find two distinct cases, corresponding to $X \approx 0$
and $F_X \approx 0$, respectively. For the case where $X\approx0$ the evolution
of $\phi$ and hence $w$ is described by only two parameters, and $w(a)$ is
model-independent and coincides with similar behavior seen in thawing
quintessence models. This result also extends to non-separable Lagrangians
where $X\approx0$. For the case $F_X \approx 0$, an expression is derived for
$w(a)$, but this expression depends on the potential $V(\phi)$, so there is no
model-independent limiting behavior. For the $X \approx 0$ case, we derive
observational constraints on the two parameters of the model, $w_0$ (the
present-day value of $w$), and the $K$, which parametrizes the curvature of the
potential. We find that the observations sharply constrain $w_0$ to be close to
-1, but provide very poor constraints on $K$.
