- Use basic rules to find limits for polynomial and rational functions
- Find function limits by breaking them into simpler parts (factoring) or using conjugates
- Determine limits by applying the squeeze theorem
Deriving the Formula for the Area of a Circle
Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287−212 BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit.
We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of [latex]n[/latex] triangles. By taking the limit as the vertex angle of these triangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: