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Limiting case analysis (LCA) is important to practicing physicists. Yet, there is little concrete guidance for physics educators, and a lack of consensus in the research community about how to help students learn, and learn from, limiting case analysis. In this study, we first review existing literature to find commonalities and variations in how instructors encourage and assess students' limiting case analysis and to highlight how it has been used by practicing physicists. Then, we examine written work from successive cohorts of physics students, all of whom have completed a course with the same instructor who emphasizes limiting case analysis in his teaching. We frame our analysis largely in terms of the theoretical framework of "adaptive expertise," finding support in the literature for the view that it is the nonalgorithmic and even playful aspects of LCA that are instrumental to its alignment with adaptive expertise rather than routine expertise. Analysis of students' commentary about how they decide which limiting cases to examine when evaluating the reasonableness of an equation provides new insights into how LCA might be better supported in the classroom so that more students can access this important tool of physics.