  The paper is dedicated to the study of embeddings of the anisotropic Besov
spaces $B^{\beta_1,...,beta_n}_{p;\theta_1,...,\theta_n}(\Bbb R^n)$ into
Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants
when some of the exponents $\beta_k$ tend to 1 ($\beta_k<1)$. In particular,
these results give an extension of the estimate proved bt\'y Bourgain, Brezis,
and Mironescu for isotropic Besov spaces. Also, in the limit, we obtain a link
with some known embeddings of anisotropic Lipschitz spaces.
  One of the key results of the paper is an anisotropic type estimate of
rearrangements in terms of partial moduli of continuity.
