Exploring Limits
The basic idea
As you will see, limits play a crucial role in defining the two main tools of calculus: the derivative and the integral. But what is a limit, anyway?
To explore this concept, let's first consider the "-gon", or regular -sided polygon. By "regular", we mean that the polygon has equal side lengths (or, equivalently, its internal angles have the same measure). The smallest of the -gons is the 3-gon or equilateral triangle. Next is the 4-gon or square, followed by the 5-gon or regular pentagon. The list goes on, and that's where limits come in. The big question is: what does this list of shapes approach, if anything? Or, as the question is more loosely stated in the applet below, "what's a regular -gon?"
As increases without bound (i.e. as it approaches ), we obtain an infinite sequence of -sided regular polygons, or . The symbol represents what this sequence of geometric shapes (or ) approaches as approaches (or ). What (geometric shape) does this sequence of geometric shapes approach? Or, more mathematically:
What is ?
More generally, a limit answers the same sort of question: where is this sequence going?
What's ?