<p>Troesch’s problem is well known for its extreme sensitivity to large parameters, which poses challenges for both numerical computation and asymptotic analysis. In this paper, we study a broader class of highly sensitive boundary-value problems generalizing the classical Troesch’s problem. We develop a refined asymptotic framework that captures the solution profiles near the boundary and establishes the emergence of boundary concentration as the parameter tends to infinity. While numerical studies have been carried out extensively, to the best of our knowledge, this appears to be the first rigorous asymptotic analysis characterizing the boundary behavior of solutions to Troesch-type problems in the large-parameter regime.</p>