  We introduce the notion of level-$\delta$ limit linear series, which describe
limits of linear series along families of smooth curves degenerating to a
singular curve $X$. We treat here only the simplest case where $X$ is the union
of two smooth components meeting transversely at a point $P$. The integer
$\delta$ stands for the singularity degree of the total space of the
degeneration at $P$. If the total space is regular, we get level-1 limit linear
series, which are precisely those introduced by Osserman in 2006. We construct
a projective moduli space $G^r_{d,\delta}(X)$ parameterizing level-$\delta$
limit linear series of rank $r$ and degree $d$ on $X$, and show that it is a
new compactification, for each $\delta$, of the moduli space of Osserman exact
limit linear series, an open subscheme $G^{r,*}_{d,1}(X)$ of the space
$G^r_{d,1}(X)$ already constructed by Osserman. Finally, we generalize work by
Esteves and Osserman by associating to each exact level-$\delta$ limit linear
series $\mathfrak g$ on $X$ a closed subscheme $\mathbb P(\mathfrak g)\subseteq
X^{(d)}$ of the $d$th symmetric product of $X$, and showing that $\mathbb
P(\mathfrak g)$ is the limit of the spaces of divisors associated to linear
series on smooth curves degenerating to $\mathfrak g$ on $X$, if such
degenerations exist. In particular, we describe completely limits of divisors
along degenerations to such a curve $X$.

Algebraic Geometry