The concept of the limit is often experienced as a very abstract concept. In this study we explore the understanding of the concept of the limit with 74 students who are prospective high school mathematics teachers. The study is located within the first of three calculus courses which they study as part of their bachelor's degree in education at a university in South Africa. The students were asked to describe what they understood by the limit of a function at a certain point. The written responses were analysed inductively and six categories were identified as descriptors of the responses. Understanding the concept of the limit of a function requires the coordination of many processes, encompassing various constructs which must satisfy certain conditions for the limit to exist. The students in each category seemed to have differentiated between some constructs and conditions but struggled with others. Their responses also revealed differences in the way in which the language was used to describe the concept. This study strengthens existing research findings about students' struggles with the formal mathematical language as well as their struggles with dealing with variation induced by the different constructs embedded in the concept.