TITLE: Lengths over a local ring
QUESTION [12 upvotes]: Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$.
Is the following claim true?
Claim:
For any $\epsilon>0$, there exists a positive integer $n$ s.t. for any ideal $I$ satisfying
1) $ I \subset\mathfrak{m^n}$
2) $\sqrt  I = \mathfrak{m}$
3) $I$ can be generated by $d$ elements,
the following holds: 
$$ \mbox{length}(A/(I+As)) /\mbox{length}(A/I) < \epsilon$$
Note: The following example shows that the claim can be false if one drops the requirement that that the number of generators of $I$ be bounded. 
Example: $A:= k[x,s]$, and let $\mathfrak{m}$ denote the ideal $(x,s)$. Let $I_{n,m}$ be an ideal of $A$ given by 
$$ I_{n,m}= s\mathfrak{m}^{n-1} + \mathfrak{m}^m$$.
We can calculate that for any $n$, 
$$\lim_{m\to \infty}  \mbox{length}(A/(I_{n,m}+As)) /\mbox{length}(A/I_{n,m}) = 1$$

REPLY [4 votes]: Here is a counterexample for you question.
Let $A = k[[s, x]]$, $\dim A = 2$
For each pair $n, m$, $n < m$, we consider the parameter ideal
$$\mathfrak{q}_{n, m} = (s^n+x^m, sx^{n-1})$$
We have $\mathfrak{q}_{n, m} + sA = (s, x^m)$. Hence 
$$\ell(A/(\mathfrak{q}_{n, m} + sA)) = m$$
On the other hand, we can check that $s^{n+1}$ and $x^{m+n-1}$ is contained in $\mathfrak{q}_{n, m}$. Thus
$$\ell(A/\mathfrak{q}_{n, m}) \leq \ell(A/(s^{n+1}, sx^{n-1},x^{m+n-1})) = m + n^2-1.$$
Therefore
$$\lim_{m \to \infty} \ell(A/(\mathfrak{q}_{n, m} + sA))/ \ell(A/\mathfrak{q}_{n, m}) = 1$$
Remark:
(i) It should be noted that, I contruct this example based thinking the minimal reduction of the ideal $I_{n,m}$ of your question.
(ii) Your question is true in the case $I = \mathfrak{m}^n$, it means
$$\lim_n \;\ell(A/(\mathfrak{m}^n + sA))/ \ell(A/(\mathfrak{m}^n) = 0.$$