The purpose of this research was to investigate student conceptions of the topic of asymptotes of rational functions and to understand the connections that students developed between the closely related notions of asymptotes, continuity, and limits. The participants of the study were university students taking Calculus 2 and were mostly freshmen. The study of rational functions and asymptotes follows the study of functions in College Algebra. The function concept is a fundamental topic in the field of mathematics and physical sciences. The concept of asymptotes is closely related to the concepts of limits, continuity, and indeterminate forms in Calculus 1. Therefore, investigating student beliefs of asymptotes and the connections with related topics could possibly shed light onto effective ways of instructing of these concepts. Qualitative methodology was used to conduct this investigation. The investigation was conducted through two problem-solving interviews and several teaching episodes. The goal of each problem-solving interview was to gain an in depth understanding of students' thinking processes while solving problems. Nineteen Calculus 2 students participated in the first problem solving interview that investigated student concept images of asymptotes of rational functions and the connections students have developed among these concepts. The interview was about two hours long. Based on the results of this interview, eight students were selected, and seven students completed teaching episodes. The teaching episodes lasted for one hour and thirty minutes and were conducted twice a week for four weeks. The participants of the teaching episodes were divided into two groups; one group consisted of 4 students, and the other group consisted of 3 students. Thus, adjustments could me made in the teaching episodes of the second group based on the observations of the first group. The purpose of these teaching episodes was to create a model of student thinking while they re-construct and re-configure the misconceptions revealed during the first interview. Through the teaching experiment the researcher was be able to gain first-hand insights into students' mathematical reasoning (Steffe & Thompson, 2000). This was accomplished by closely observing students' problem solving procedures, listening to their discussions, and taking notes during the entire process. Data were collected through students' written work, the researcher comments and field notes, and the transcripts of the videotapes of the interviews as well as teaching episodes. After the conclusion of the teaching episodes, the second problem-solving interview was conducted with seven students and videotaped. These semi-structured interviews were to determine their concept images after the activities of the teaching episodes and to create models of student conceptions. [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.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]