The purpose of this study is to investigate routines as guides for mathematical thinking. Four English-speaking and four Korean-speaking students were interviewed in English about the concepts of limit and infinity. Based on the communicational approach to cognition, which views mathematics as a discourse, we identified the primary characteristics of students' routines for infinity and limit. Results show that language differences between English and Korean affect students' problem-solving routines in mathematics. On the basis of these results, we conclude that there is a need to deal with linguistic sensitivity in mathematics learning.