An Empirical Study of Student Errors in Solving Absolute Value Inequalities and Function Limits

Abstract

Solving absolute value inequalities and proving limits are foundational competencies in early calculus. However, many prospective mathematics teachers rely on procedural routines that fail when algebraic constraints and formal justification are required. This mixed-methods research combines a descriptive tally of written errors with follow-up clinical interviews to examine students’ error patterns and the factors underlying them. Participants were 25 second-semester calculus students in a Mathematics Education program. A test on absolute value inequalities and limits (including an item requiring avoidance of division by zero and an ε–δ proof) was adapted from Purcell and Varberg and reviewed by three calculus lecturers for content relevance and clarity. Results indicate that routine limit evaluation by direct substitution was generally successful; however, performance dropped sharply for tasks demanding structural reasoning: no participant solved the limit item involving a removable discontinuity with a domain restriction, and only one participant produced an acceptable ε–δ argument. Across tasks, errors clustered into (i) conceptual errors (misinterpreting absolute value as “always positive” and conflating limit value with function value), (ii) systematic procedural errors (incorrect case-splitting, invalid algebraic transformations, and unjustified cancellation), and (iii) random/careless errors. Interview data suggest that incomplete concept images, fragile prerequisite algebra, and low confidence when facing proof-oriented prompts jointly contributed to these errors. The findings support instruction that explicitly foregrounds domain restrictions, connects multiple representations of limit, and scaffolds ε–δ reasoning through guided bounding and δ(ε) construction