G. Olshanski calls the harmonic analysis of noncommutative groups with infinite dimensional dual space an important "chapter of representation theory". And some of the main objects you see in this chapter are the infinite symmetric group $S(\infty)$, the infinite bisymmetric group $G=S(\infty)\times S(\infty)$ and the space of virtual permutations $\mathfrak{S}$, which is a compactification of $S(\infty)$ (It is not a group but it is a $G$-space).

$S(\infty)$ is the group of all finite permutations of $\mathbb N$. In other words, since each finite symmetric group $S_n$ acts on $[n]=\{1,2,\dots,n\}$, and the stabilizer of $n$ is canonically isomorphic to $S_{n-1}$ you get an embedding $S_{n-1}\to S_n$ and define $S(\infty)$ as the direct limit with respect to these embeddings.

Similarly you can define projections $S_n\to S_{n-1}$ by removing $n$ from the cycle that contains it and take the projective limit. This will give you $\mathfrak S$, which is equipped with the projective limit topology, and is a totally disconnected compact topological space. The haar measure of $S_n$ passes on to $\mathfrak S$ and is the unique measure invariant under the action of $G$. This is just an introduction to what you can find in this survey.

I felt like mentioning this because you didn't define what you meant by limit or permutation in your question, and I am giving a possible answer in terms of objects that appear frequently in literature. If this is satisfactory to you then the answer to your second question is no, because the image of $S(\infty)$ is dense in $\mathfrak S$. Note that one can still talk about properties of "permutations" in $\mathfrak S$, such as the distribution of cycle lengths etc.

G. Olshanski calls the harmonic analysis of noncommutative groups with infinite dimensional dual space an important "chapter of representation theory". And some of the main objects you see in this chapter are the infinite symmetric group $S(\infty)$, the infinite bisymmetric group $G=S(\infty)\times S(\infty)$ and the space of virtual permutations $\mathfrak{S}$, which is a compactification of $S(\infty)$ (It is not a group but it is a $G$-space).

$S(\infty)$ is the group of all finite permutations of $\mathbb N$. In other words, since each finite symmetric group $S_n$ acts on $[n]=\{1,2,\dots,n\}$, and the stabilizer of $n$ is canonically isomorphic to $S_{n-1}$ you get an embedding $S_{n-1}\to S_n$ and define $S(\infty)$ as the direct limit with respect to these embeddings.

Similarly you can define projections $S_n\to S_{n-1}$ by removing $n$ from the cycle that contains it and take the projective limit. This will give you $\mathfrak S$, which is equipped with the projective limit topology, and is a totally disconnected compact topological space. The Haar measure of $S_n$ passes on to $\mathfrak S$ and is the unique measure invariant under the action of $G$. This is just an introduction to what you can find in this survey.

I felt like mentioning this because you didn't define what you meant by limit or permutation in your question, and I am giving a possible answer in terms of objects that appear frequently in literature. If this is satisfactory to you then the answer to your second question is no, because the image of $S(\infty)$ is dense in $\mathfrak S$. Note that one can still talk about properties of "permutations" in $\mathfrak S$, such as the distribution of cycle lengths etc.