This study investigated and compared how native-English and native-Korean speaking university students, who received their education respectively in the U.S. and in Korea, thought about the concepts of "infinity" and "limit". The primary motivation for this study was the discontinuity in Korean and the continuity in English between the non-mathematical and mathematical discourses on "infinity" and "limit": non-mathematical uses of the mathematical words infinity and limit appear only in English. Based on the communicational approach to cognition, according to which mathematics is a kind of discourse, the characteristics of students' discourse on the topics were identified. The participants' discourse was scrutinized with an eye to the common characteristics as well as culture- and education-related differences. The setting for the study was a calculus class for university students in the U.S. and Korea. Methodology involved surveys and interviews. A total of 132 English speakers and 126 Korean speakers participated in the survey. Within each linguistically distinct group, twenty paired representatives were selected from the survey participants for follow-up interviews. The detailed discourse analyses of the interview transcripts generated preliminary hypotheses in the interviewees' discourse on "infinity" and "limit". Overall analyses of response patterns involved searching for frequencies and percentages of the survey participants' responses and comparing the proportions of the two groups through chi-square analysis to confirm the emerging hypotheses. Data from both sources were used to elaborate and verify hypotheses. In order to attribute the specific source of differences in mathematical discourses of the two groups, students' backgrounds which were likely to influence their mathematical discourses on "infinity" and "limit" were also investigated in this study. It is concluded from this study that the mathematical discourse of U.S. English speakers seemed to be growing continuously from their non-mathematical discourse, but there was no such continuity in the case of Korean speakers. The use of the mathematical nouns "infinity" and "limit" by English speakers was processual, just as it was in their non-mathematical, colloquial uses, whereas Korean speakers' language was more formally mathematical and structural. This shows in developing their discourses on "infinity" and "limit", English speakers seemed to build more on their previous everyday experience with discourses on "infinity" and "limit", whereas Korean speakers had no choice but to rely on the structural discourse of mathematical textbooks. These differences in the use of the words "infinity" and "limit" between U.S. and Korean groups seemed to result in differences in endorsed narratives accepted by the students as true, "routines" they used in problem solving, and "visual mediators" they applied. In general, none of the mathematical discourses, either of English or Korean speakers, was fully compatible with the canonical mathematical discourse. The results of this study provide strong support for the conjecture that non-mathematical discourse shapes mathematical discourse of students--a fact that should be kept in mind in both the research on advanced mathematical concepts and teaching advanced mathematics. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]