This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students' existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking. Through exploration of the monotonic convergence of a sequence to a finite limit and oscillatory convergence of a sequence to a finite limit, the two examples recast the essential mathematics in a way that allows liberal arts students to develop a mathematically correct appreciation for convergence to the limit. (Contains 9 figures and 1 table.)