What can the approximation of a group by some class be used for? Recall the following concept due to Malcev and Gromov. Let $C$ be some class of groups. A group $G$ is said to be approximable by the class $C$ if for every finite symmetric subset $F\subset G$ containing 1, there is a group $H$ in $C$ and an injection $f:F\to H$ such that $f(1)=1$ and $f(x)f(y)=f(z)$ for every $x,y,z\in F$ with $xy=z$.
My question is: why is this concept useful? I.e. what are real application of it in group theory (say, for establishing some properties of $G$ via its approximation by some class)?
 A: Let $\mathcal{F}_k$ be the class of free groups that can be generated by at most $k$ elements.  In your terminology, limit groups are the groups that can be approximated by the groups in $\mathcal{F}_k$.  This article by Champetier--Guirardel develops the theory of limit groups from this point of view. It contains many examples of the kinds of applications that you're looking for.
A: In general approximation or continuity results for various invariants and functions might be used to extend results from the class $C$ to the class of $C$-approximable groups. 
A concrete example is the Determinant Conjecture of W. Lück : it states that for a group $G$ and a matrix $A$ over the group-ring $\mathbb ZG$ its Fuglede--Kadison determinant $\det_{\mathcal NG}(A)$ is $\ge 1$. This has applications to algebraic topology. (For details about this see Lück's book $L^2$-invariants : theory and applications to geometry and $K$-theory, especially Chapter 13.) Note that in general it is not known whether $\det_{\mathcal NG}(A) > 0$. 
This conjecture holds for all residually finite groups, and this is established as follows : it is trivial for $G$ finite where the Fuglede--Kadison determinant is essentially the product of singular values (eigenvalues of the symmetrised matrix). Then one proves that in a sequence of finite quotients $G_n$ approximating (sofically) $G$, if the matrices $A_n$ are the images of $A$ we have 
$$ \limsup_{n\to +\infty} \det{}_{\mathcal NG_n} \le \det{}_{\mathcal NG}(A) $$
and the result follows immediately (this is written up in detail in loc. cit.). I think the argument immediately extends to any sofic approximation though I'm not sure if the details are written up somewhere. 
