Analysis is one of the large divisions of modern mathematics.
Analysis is the study of limits .
Anything in mathematics which has limits in it uses ideas of analysis. Some of the examples which will be important in this course are sequences, infinite series, derivatives of functions, and integrals. As you know from calculus, limits are the basis of understanding integration and differentiation, and, as you also know from calculus, these things are the basis of everything in the world you could ever need to know. Probably, when you took calculus, you spent a lot of time learning techniques of how to integrate and differentiate, and actually learning about limits was dealt with in a fairly short time. But limits are a very rich, and very subtle concept.