  It is shown that the difference equation \begin{equation}\label{abseq}
(\Delta f(z))^2=A(z)(f(z)f(z+1)-B(z)), \qquad\qquad (1) \end{equation} where
$A(z)$ and $B(z)$ are meromorphic functions, possesses a continuous limit to
the differential equation \begin{equation}\label{abseq2}
(w')^2=A(z)(w^2-1),\qquad\qquad (2) \end{equation} which extends to solutions
in certain cases. In addition, if (1) possesses two distinct transcendental
meromorphic solutions, it is shown that these solutions satisfy an algebraic
relation, and that their growth behaviors are almost same in the sense of
Nevanlinna under some conditions. Examples are given to discuss the sharpness
of the results obtained. These properties are counterparts of the corresponding
results on the algebraic differential equation (2).
