Identification of input configurations so as to meet a prespecified output target under a limited experimental budget has been an important task for computer experiments. Such a task often involves the development of response models and design of experimental trials that rely on the models exhibiting continuity and differentiability properties. Motivated by two canonical examples in systems and manufacturing engineering, we propose a strategy for locating the boundary of the response surface in computer experiments, wherein on one side the response is finite, whereas on the other side it is infinite, leveraging ideas from active learning and quasi-Monte Carlo methods. The strategy is illustrated on an example from computer networks engineering and one from precision manufacturing and shown to allocate experimental trials in a fairly effective manner. We conclude by discussing extensions of the proposed strategy to characterize other types of output discontinuity or nondifferentiability in high-cost experiments, including jump discontinuities in the target output response or pathological structures such as kinks and cusps.