  Singular exponential nonlinearities of the form $e^{h(x)\epsilon^{-1}}$ with
$\epsilon>0$ small occur in many different applications. These terms have
essential singularities for $\epsilon=0$ leading to very different behaviour
depending on the sign of $h$. In this paper, we consider two prototypical
singularly perturbed oscillators with such exponential nonlinearities. We apply
a suitable normalization for both systems such that the $\epsilon\rightarrow 0$
limit is a piecewise smooth system. The convergence to this nonsmooth system is
exponential due to the nonlinearities we study. By working on the two model
systems we use a blow-up approach to demonstrate that this exponential
convergence can be harmless in some cases while in other scenarios it can lead
to further degeneracies. For our second model system, we deal with such
degeneracies due to exponentially small terms by extending the space dimension,
following the approach in Kristiansen [20], and prove - for both systems -
existence of (unique) limit cycles by perturbing away from singular cycles
having desirable hyperbolicity properties.
