  Given a Brownian Motion $W$, in this paper we study the asymptotic behavior,
as $\eps \to 0$, of the quadratic covariation between $f (\eps W)$ and $W$ in
the case in which $f$ is not smooth. Among the main features discovered is that
the speed of the decay in the case $f \in C^\alpha$ is polynomial in $\eps$ and
not exponential as expected. We use a recent representation as a backward-
forward It\^o integral of $[f (\eps W), W]$ to prove an $\eps$-dependent
approximation scheme which is of independent interest. We get the result by
providing estimates to this approximation.
