  This paper begins to study the limiting behavior of a family of Hermitian
Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector
bundles over a product of two elliptic curves with K\"ahler metrics
$\omega_\epsilon$ when $\epsilon\to 0$. Here $\omega_\epsilon$ are flat and
have areas $\epsilon$ and $\epsilon^{-1}$ on the two elliptic curves
respectively. A family of Hermitian metrics on the vector bundle are explicitly
constructed and with respect to them, the HYM metrics are normalized. We then
compare the family of normalized HYM metrics with the family of constructed
Hermitian metrics by doing estimates. We get the higher order estimates as long
as the $C^0$-estimate is provided. We also get the estimate of the lower bound
of the $C^0$-norm. If the desired estimate of the upper bound of the $C^0$-norm
can be obtained, then it would be shown that these two families of metrics are
close to arbitrary order in $\epsilon$ in any $C^k$ norms.
