My question concerns the following situation: Let $G$ be a domain in $\mathbb{C}$ and $f_n: G \rightarrow \mathbb{C}$ be a sequence of quasiconformal mappings. Suppose that $f_n$ converges uniformly on $G$, i.e. with respect to the supremum metric on $G$, to a homeomorphism $f: G \rightarrow \mathbb{C}$. My question is:
Is it possible that the limit mapping $f$ is NOT quasiconformal, even though being a homeomorphism?
Note that the answer is surely "NO" if the $f_n$ are all $K$-quasiconformal for some fixed $K < \infty$, by well-known convergence results on quasiconformal mappings; in this case, the limit mapping $f$ will be $K$-quasiconformal again. Hence the interesting part of my question is when the condition "the $f_n$ are $K$-quasiconformal mappings" is dropped, i.e. the maximal dilatations of the $f_n$ are not uniformly bounded by some constant $K$...I do not know what could possibly happen in this situation (maybe the situation at hand actually forces the $f_n$ to be $K$-quasiconformal?), unfortunately all convergence results on quasiconformal mappings I am aware of deal with the situation that the $f_n$ are all $K$-quasiconformal. Any kind of help is highly appreciated - thanks in advance!