  For a graph $H$, let $$c_{\infty}(H)= \lim_{n \to
\infty}\max\frac{|E(G)|}{n},$$ where the maximum is taken over all graphs $G$
on $n$ vertices not containing $H$ as a minor. Thus $c_{\infty}(H)$ is the
asymptotic maximum density of graphs not containing $H$ as a minor. Employing a
structural lemma due to Eppstein, we prove new upper bounds on $c_{\infty}(H)$
for disconnected graphs $H$. In particular, we determine $c_{\infty}(H)$
whenever $H$ is union of cycles. Finally, we investigate the behaviour of
$c_\infty(sK_r)$ for fixed $r$, where $sK_r$ denotes the union of $s$ disjoint
copies of the complete graph on $r$ vertices. Improving on a result of
Thomason, we show that $$c_\infty(sK_r)=s(r-1)-1 \mathrm{\; for \;} s
={\Omega}\left(\frac{\log{r}}{\log\log{r}}\right),$$ and
$$c_\infty(sK_r)>s(r-1)-1 \mathrm{\; for \;} s
={o}\left(\frac{\log{r}}{\log\log{r}}\right).$$
