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)?