  It is well known that the Tits boundary of a proper cocompact CAT(0) space
embeds into every asymptotic cone of the space. We explore the relationships
between the asymptotic cones of a CAT(0) space and its boundary under both the
standard visual (i.e. cone) topology and the Tits metric. We show that the set
of asymptotic cones of a proper cocompact CAT(0) space admits canonical
connecting maps under which the direct limit is isometric to the Euclidean cone
on the Tits boundary. The resulting projection from any asymptotic cone to the
Tits boundary is determined by the visual topology; on the other hand, the
visual topology can be recovered from the connecting maps between asymptotic
cones. We also demonstrate how maps between asymptotic cones induce maps
between Tits boundaries.
