This paper analyses how pre-service teachers approach the notion of the infinite limit of a sequence from two perspectives: Specialized Content Knowledge and Advanced Mathematical Thinking. The aim of this study is to identify the difficulties associated with this notion and to classify them. In order to achieve this, an exploratory qualitative approach was applied using a sample of 12 future teachers. Among the results, we can affirm that pre-service teachers mainly use algorithmic procedures to solve tasks in which this type of limit is implicit, although they would consider a resolution that specifically involves the notion with an intuitive approach if they had to explain it to their students.