This study was conducted to reveal how students at university level justify their solutions to tasks with various degrees of difficulty. The study is part of a larger study of students' concept formation of limits. The mathematical area is limits of functions. The study was carried out at a Swedish university at the first level of mathematics. The results are, however, applicable to other countries as well since students meet similar challenges in their learning of limits. In discussions with some Australian mathematics teachers at university level, the author, found out that the topics taught in basic mathematics courses in Australia are similar to Swedish courses. Two groups of students taking the same course in successive semesters have been solving tasks. Their solutions are categorised and analysed to create a picture of how students reason about limits. (Contains 5 tables.)