5 narrowed the question to the special case B= dim A_m.

Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$. \mathfrak{m}$,$d= \dim A_\mathfrak{m}$. Is the following claim true? Claim: For any$B>0$and$\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$can be generated by (at most)$B$d$ 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 $n$, $$\lim_{m\to \infty} \mbox{length}(A/(I_{n,m}+As)) /\mbox{length}(A/I_{n,m}) = 1$$

4 rephrased the question so that it makes sense for any (not just local) noetherian ring

Let $A$ be a noetherian local domain, $\mathfrak{m}$ the а maximal ideal, $s$ a non-zero , non-unit element of $A$. \mathfrak{m}$. Is the following claim true? Claim: For any$B>0$and$\epsilon>0$, there exists a positive integer$n$s.t. for any parameter ideal$I$contained in$\mathfrak{m^n}$,$satisfying

1) $I \mbox{length}(A/(I+As)) /\mbox{length}(A/I) < subset\mathfrak{m^n}$

2) $\sqrt I = \epsilon$$(I is a parameter ideal if it mathfrak{m} 3) I can be generated by (at most) \dim A B elementsand , the following holds: \sqrt I = \mathfrak{m}.)mbox{length}(A/(I+As)) /\mbox{length}(A/I) < \epsilon$$ Note: regarding the requirement that$I$be a parameter ideal, the The following example shows that the claim is can be false if one puts no restriction on drops the requirement that that the number of generators of$I$other than it be$\mathfrak{m}$-primary. bounded. Example:$A:= k[x,s]$, and let$\mathfrak{m}$denote the ideal$(x,y)$. (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 $n$, $$\lim_{m\to \infty} \mbox{length}(A/(I_{n,m}+As)) /\mbox{length}(A/I_{n,m}) = 1$$

3 Add an example showing that some restriction on $I$ is necessary

Let $A$ be a noetherian local domain, $\mathfrak{m}$ the maximal ideal, $s$ a non-zero, non-unit element of $A$. Is the following claim true?

Claim: For any $\epsilon>0$, there exists a positive integer $n$ s.t. for any parameter ideal $I$ contained in $\mathfrak{m^n}$, $$\mbox{length}(A/(I+As)) /\mbox{length}(A/I) < \epsilon$$

($I$ is a parameter ideal if it can be generated by $\dim A$ elements and $\sqrt I = \mathfrak{m}$.)

Note: regarding the requirement that $I$ be a parameter ideal, the following example shows that the claim is false if one puts no restriction on $I$ other than it be $\mathfrak{m}$-primary.

Example: $A:= k[x,s]$, and let $\mathfrak{m}$ denote the ideal $(x,y)$. 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 $n$, $$\lim_{m\to \infty} \mbox{length}(A/(I_{n,m}+As)) /\mbox{length}(A/I_{n,m}) = 1$$

2 polished up and put def'n of parameter ideal
1