1. Archimedes Limits by Archimedes of Syracuse

2. There is a fascinating history about how they made an attempt to calculate the area of a circle.

3. The Greeks weren’t very good with a circle, but they were good with polygons

4. They would find the area of the square (s 2 ), and they would either ignore the light blue part, or they would add on “just a bit”… 4 sides

5. Then Archimedes had an idea… He thought of what is now known as the “method of Exhaustion” What if we inscribed (meaning the vertices of the polygon touches the edge of the circle) shapes with more sides than 4?....

6. There is less light blue area… …area that is not counted. 6 sides

7. The amount of blue space is less now… 8 sides

9. What about a decagon? How about a myriagon (10000 sides) What happens as the number of sides of the polygon approaches infinity?

10. Archimedes’ method of finding the area of a circle is based on the concept of a limit. The circle is the limiting shape of the polygon. As the number of sides gets larger, the area of the polygon approached its limit, the shape of the circle, without ever becoming an actual circle. Also, as “h” approaches zero, the slope of the secant approaches its limiting value, the slope of the tangent.

11. Archimedes not only approached the area of a circle from the inside, but from the outside as well.

12. 4 sides

13. The amount of blue space is less now… 8 sides

14. Archimedes approach can also be applied to determining the limit of a function. A limit can be approached from the left side and the right side, called left hand limits and right hand limits. To evaluate a left hand limit, we use values that are smaller than the value being approached. To evaluate right hand limits, we use values that are larger than the value being approached.

15. Try investigation “limit of a sequence” Try investigation “limit of a function from the equation”