  We consider smooth systems limiting as $\epsilon \to 0$ to piecewise-smooth
(PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable
local normal form, we study the dynamics for the smooth system with $0 <
\epsilon \ll 1$ using a combination of geometric singular perturbation theory
and blow-up. We show that the type of BF bifurcation in the PWS system
determines the bifurcation structure for the smooth system within an
$\epsilon-$dependent domain which shrinks to zero as $\epsilon \to 0$,
identifying a supercritical Andronov-Hopf bifurcation in one case, and a
supercritical Bogdanov-Takens bifurcation in two other cases. We also show that
PWS cycles associated with BF bifurcations persist as relaxation cycles in the
smooth system, and prove existence of a family of stable limit cycles which
connects the relaxation cycles to regular cycles within the
$\epsilon-$dependent domain described above. Our results are applied to models
for Gause predator-prey interaction and mechanical oscillation subject to
friction.
