Constructive Mathematics might be regarded 
as a fragment of classical mathematics in 
which any proof of an existence theorem 
is equipped with a computable function 
giving the solution of the theorem.  
Limit Computable Mathematics (LCM) 
considered in this note is a fragment 
of classical mathematics in which 
any proof of an existence theorem is 
equipped with a function computing 
the solution of the theorem {\em in 
the limit}.

Computation in the limit, or more 
formally, limiting recursion, is a central 
notion of learning theory by Gold and 
Putnam \cite{Gold1, Putnam, STL}. 
We will show that a realizability
interpretation via limiting recursive 
functions is a natural modeling
of LCM for first order arithmetic.
\par
We will point out that this will 
enable automatic extraction of 
{\it limit-algorithms} from some 
classical proofs of well-known 
transfinite theorems, e.g., Hilbert's 
original proof of his famous finite 
basis theorem, once blamed as 
``theology'' by P. Gordan.