Physicists seek to understand the world and often use mathematics to aid in attaining that goal. Thus when doing mathematics they often interrogate their results in order to ensure that they align with their expectations and to gain new insight into how the world works. This seeking of coherence between the multiple ways of understanding the world is called physics sensemaking. One way physicists do sensemaking is to perform a special-case analysis: they set a parameter of the problem to a specific value that corresponds to a situation - a case - where they know or have intuition about the result, and then make a judgment about the correctness of their answer. In this research project, I examine when and how physics students engage with sensemaking while solving physics problems, with particular interest in their use of special-case analysis. In the first study, an analysis of homework from students in a sophomore mechanics class with an explicit emphasis on physics sensemaking showed that students engaged with many sensemaking strategies, with special-case analysis being the most common way they checked their answers. This original research led to a more detailed study of how students perform special-case analysis. This second study analyzes homework problems with three different types of prompts: one where students were explicitly prompted to perform a special-case analysis for a specific case, one where students were prompted to perform special-case analysis but were free to choose their case, and one where students were asked to do some sensemaking on their answer but special-case analysis was not specified. We found that students used a variety of types of reasoning to defend and understand the special-cases they analyzed. From both studies we found that some student reasoning was expert-like. However, students often applied atypical sensemaking strategies or used reasoning like "correct mathematics" and descriptions of the behavior of functions to reconcile their special-case analysis. This project, as a whole, articulates the kinds of sensemaking students chose to engage with and the specific reasoning patterns students use most frequently when performing special-case analysis.