Limits and Continuity

The most fundamental building block of calculus is the concept of a limit. They are the key to understanding everything in calculus, from derivatives to integrals. As such, it is crucial to have a solid understanding of what limits are and how they work.

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It is interesting to note, however, that the concept of a limit was not rigorously defined until the 19th century. In fact, most of calculus, such as derivatives and integrals, was developed without the concept of limits. Furthermore, the order in which calculus is taught today is not the order in which it was developed. It's pretty much the opposite; courses in calculus today typically start with limits, then move on to derivatives, and finally integrals. However, the historical development of calculus was the opposite: it started with integrals, then moved on to derivatives, and finally limits. This means that (1) the historical development of calculus was not as rigorous as it is today, and (2) we do not actually "need" a formal treatment of limits to understand ideas in calculus.