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# Chapter 1: Limits

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## Limits

Two key problems led to the initial formulation of calculus:

1. the tangent problem, or how to determine the slope of a line tangent to a curve at a point, and
2. the area problem, or how to determine the area under a curve.

The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach.