  The planar visible fold is a simple singularity in piecewise smooth systems.
In this paper, we consider singularly perturbed systems that limit to this
piecewise smooth bifurcation as the singular perturbation parameter
$\epsilon\rightarrow 0$. Alternatively, these singularly perturbed systems can
be thought of as regularizations of their piecewise counterparts. The main
contribution of the paper is to demonstrate the use of consecutive blowup
transformations in this setting, allowing us to obtain detailed information
about a transition map near the fold under very general assumptions. We apply
this information to prove, for the first time, the existence of a locally
unique saddle-node bifurcation in the case where a limit cycle, in the singular
limit $\epsilon\rightarrow 0$, grazes the discontinuity set. We apply this
result to a mass-spring system on a moving belt described by a Stribeck-type
friction law.
