  Let F be the (Thompson's) group < x_0, x_1 | [x_0x_1^-1, x_0^-ix_1 x_0^i],
i=1,2 >. We study the structure of F-limit groups. Let G_n= < y_1,..., y_m,
x_0,x_1 | [x_0x_1^-1,x_0^-1x_1x_0],[x_0x_1^-1,x_0^-2x_1x_0^2],
y_j^-1g_j,n(x_0,x_1), 0<j<m+1 >, where g_j,n(x_0,x_1) belongs to F, be a family
of groups marked by m+2 elements. If the sequence (G_n)_n<w is convergent in
the space of marked groups and G is the corresponding limit we say that G is an
F-limit group. Primarily the paper is devoted to the study of F-limit groups.
  The results are based on some theorems concerning laws with parameters in F.
In particular several constructions of such laws are given. On the other hand
we formulate some very general conditions on words with parameters
w(y,a_1,...,a_n) over F which guarantee that the inequality w(y,a') is not
equal to 1 has a solution in F.
  Some of the results are of a more general nature and can be applied to study
limit groups with respect to other finitely generated groups and classes of
finitely generated groups, in particular to the case of the Grigorchuk group.
