  In this work we show two results about approximating, with respect to the
compact-open topology, mapping classes on surfaces of infinite-type by
quasi-conformal maps, in particular we are interested in density results. The
first result is that given any infinite-type surface $S$ there exists a
hyperbolic structure $X$ on $S$ such that $\text{PMCG}(S)\subseteq
\overline{\text{Mod}(X)}$, for $\text{Mod}(X)$ the set of quasi-conformal
homeomorphism on $X$. The second result is that given any surface $S$ with
countably many ends then there exists a hyperbolic structure $X$ such that
$\text{MCG} (S)=\overline{\text{Mod}(X)}$.
