In this module, we formalize some ideas that are usually introduced in precalculus. For example, most students are familiar with the idea of a continuous function. Continuous functions are usually introduced as functions you can draw without picking up your pencil. This definition should not feel very rigorous, and indeed, as you get introduced to more complicated functions, this definition will not meet your needs. Limits will be the tool needed to define continuity precisely. Similarly, horizontal and vertical asymptotes, which are often presented simply as dotted lines on the graph that the function gets close to but does not touch, are more rigorously defined. We'll also define our main object of study, the derivative of a function. And we'll do this using, you guessed it, limits. As you work through this module, realize that everything you're working with is a limit. Limits are the heart of calculus. You now working through deep ideas from the great minds of Isaac Newton and Gottfried Leibnitz. Go slowly. Make sure you understand how limits and functions interact with each other. We will continue to explore this relationship in future chapters, but it's imperative now to build a strong foundation. Let's begin. [MUSIC]