In this article, attempts were made to examine students' thinking about the concepts of infinity and their ideas about transiting from finite to infinite states through the concept of limits of sequences. The participants included 78 senior high-school students ranging in age between 17 and 19 years old. The data were collected through a questionnaire and an interview with all of the subjects. The findings showed that the students' understanding of infinity is related to finite situations and many students consider infinite processes as a generalized form of finite processes. In the present study, the most common mistakes committed by students were related to consideration of infinity as a number and application of known finite results to infinite states.