Interpreting the ε–δ Definition: Hermeneutic Insights into Students’ Ways of Thinking

Abstract

This study aims to reveal the hermeneutic insight in the formation of students’ ways of thinking (WoT) toward the precise meaning of the limit concept in differential calculus. Formally, the concept of limit is defined through the relationship between ε (epsilon) and δ (delta), namely that for every ε > 0, there exists a δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε, expressing a precise dependency between function values and domain values. However, students often interpret this definition in ways that do not fully capture its relational meaning. This qualitative case study involved 28 undergraduate students and used written tasks, interviews, and classroom observations to collect data. The analysis integrates the WoT framework, which consists of empirical, procedural, and theoretical ways of thinking, with a hermeneutic perspective to explore how students construct meaning. The findings show that most students’ reasoning is dominated by empirical and procedural ways of thinking of the ε–δ, reflected in substitution, approximation, and symbolic manipulation. From a hermeneutic perspective, these patterns indicate that students interpret limit within restricted interpretative horizons. Only a few students demonstrate theoretical reasoning, suggesting an emerging integration between prior understanding and formal structure. These findings suggest that students’ difficulties with limits are not only cognitive but also interpretative, highlighting the importance of supporting meaning-making in calculus learning.