Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the other two main branches, "algebra" and "geometry", which do not seem to have other unrelated meanings.
There is a tradition on early modern mathematics regarding the usage of the term analysis :
The background is the "rediscovery" of ancient Greek mathematics and, in particular of Pappus of Alexandria, (c.A.D. 290 – c.350) and his main work in eight books titled Synagoge or Collection, which Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem.
See Henk Bos, Redefining Geometrical Exactness. Descartes' Transformation of the Early Modern Concept of Construction (2001), page :
Reference to Pappus' problems is also found into René Descartes' La Géométrie (1637).
The two main line to be understood are :
Both, I think, are "involved" into the use of analysis to name the new method introduced by Newton and Leibniz.
You can see :
The first occurrence (1696) of the term "analysis" in the sense of the mathematical discipline extending calculus occurs in the title of l'Hopital's work Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. The term had been used earlier as part of a dichotomy analysis/synthesis, for example in Fermat. However l'Hopital was the first to use the term to describe the new science being created by Leibniz and others in the 17th century.