Summary: It is shown that differential operators arising from boundary value problems with eigenvalue parameter in the boundary condition occur as the limits, in the sense of a generalized notion of strong resolvent convergence, of families of Sturm-Liouville operators modeling heat flow in a rod, where the diffusion coefficient becomes arbitrarily large in half of the rod, thus modeling a mixing-diffusion problem. The generalized notion of strong resolvent convergence is defined, and a development is given in the setting of the abstract theory of self- adjoint operators in a Hilbert space and the theory of semigroups.