  Motivated by the picture of mirror symmetry suggested by Strominger, Yau and
Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of
Calabi-Yau n-folds (with Ricci-flat K\"ahler metric) as one approaches a large
complex structure limit point in moduli; a similar conjecture was made
independently by Kontsevich, Soibelman and Todorov. Roughly stated, the
conjecture says that, if the metrics are normalized to have constant diameter,
then this limit is the base of the conjectural special lagrangian torus
fibration associated with the large complex structure limit, namely an
n-sphere, and that the metric on this S^n is induced from a standard (singular)
Riemannian metric on the base, the singularities of the metric corresponding to
the discriminant locus of the fibration.
  This conjecture is trivially true for elliptic curves; in this paper we prove
it in the case of K3 surfaces. Using the standard description of mirror
symmetry for K3 surfaces and the hyperk\"ahler rotation trick, we reduce the
problem to that of studying K\"ahler degenerations of elliptic K3 surfaces,
with the K\"ahler class approaching the wall of the K\"ahler cone corresponding
to the fibration and the volume normalized to be one. Here we are able to write
down a remarkably accurate approximation to the Ricci-flat metric -- if the
elliptic fibres are of area $\epsilon >0$, then the error is
$O(e^{-C/\epsilon})$ for some constant $C>0$. This metric is obtained by gluing
together a semi-flat metric on the smooth part of the fibration with suitable
Ooguri-Vafa metrics near the singular fibres. For small $\epsilon$, this is a
sufficiently good approximation that the above conjecture is then an easy
consequence.
