  We determine the asymptotic behaviour of extremal length along arbitrary
Teichm\"uller rays. This allows us to calculate the endpoint in the
Gardiner-Masur boundary of any Teichm\"uller ray. We give a proof that this
compactification is the same as the horofunction compactification. An important
subset of the latter is the set of Busemann points. We show that the Busemann
points are exactly the limits of the Teichm\"uller rays, and we give a
necessary and sufficient condition for a sequence of Busemann points to
converge to a Busemann point. Finally, we determine the detour metric on the
boundary.
