  We consider the metric space of all toric K\"ahler metrics on a compact toric
manifold; when "looking at it from infinity" (following Gromov), we obtain the
tangent cone at infinity, which is parametrized by equivalence classes of
complete geodesics. In the present paper, we study the associated limit for the
family of metrics on the toric variety, its quantization, and degeneration of
generic divisors.
  The limits of the corresponding K\"ahler polarizations become degenerate
along the Lagrangian fibration defined by the moment map. This allows us to
interpolate continuously between geometric quantizations in the holomorphic and
real polarizations and show that the monomial holomorphic sections of the
prequantum bundle converge to Dirac delta distributions supported on
Bohr-Sommerfeld fibers.
  In the second part, we use these families of toric metric degenerations to
study the limit of compact hypersurface amoebas and show that in Legendre
transformed variables they are described by tropical amoebas. We believe that
our approach gives a different, complementary, perspective on the relation
between complex algebraic geometry and tropical geometry.
