  The cut-off mass ratio is under debate for contact binaries. In this paper,
we present the investigation of two contact binaries with mass ratios close to
the low mass ratio limit. It is found that the mass ratios of VSX
J082700.8+462850 (hereafter J082700) and 1SWASP J132829.37+555246.1 (hereafter
J132829) are both less than 0.1 ($q\sim0.055$ for J082700, and $q\sim0.089$ for
J132829). J082700 is a shallow contact binary with a contact degree of
$\sim$19%, and J132829 is a deep contact system with a fillout factor of
$\sim$70%. The $O-C$ diagram analysis indicated that both the two systems
manifest long-term period decrease. In addition, J082700 exhibits a cyclic
modulation which is more likely resulted from Applegate mechanism. In order to
explore the properties of extremely low mass ratio contact binaries (ELMRCBs),
we carried out a statistical analysis on contact binaries with mass ratios of
$q\lesssim0.1$ and discovered that the values of $J_{spin}/J_{orb}$ of three
systems are greater than 1/3. Two possible explanations can interpret this
phenomenon. One is that some physical processes, unknown to date, are not
considered when Hut presented the dynamically instability criterion. The other
is that the dimensionless gyration radius ($k$) should be smaller than the
value we used ($k^2=0.06$). We also found that the formation of ELMRCBs
possibly has two channels. The study of evolutionary states of ELMRCBs reveals
that their evolutionary states are similar with those of normal W UMa contact
binaries.
