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Calculating Limits

Limits Assignment / Homework Help
Limits have a very important role in developing major topics in calculus.

Let us take a look at an example using the concept of limits.

The area of a polygon can be found by dividing the shape into triangles and then finding the sum of the areas of the triangles.

But how about the area of a curved figure, such as a circle, which can't be divided into triangles?

This can very well be estimated by finding the areas of the circumscribing and inscribing polygons as shown in the figures below. The area of the circle is less than the area of the green circumscribed polygon, and greater than the area of the red inscribed polygon.

Fig. 1 Fig. 2 Fig. 3
Limits Assignment / Homework Help Limits Assignment / Homework Help Limits Assignment / Homework Help

Let the number of sides of the polygon be n, Area of inscribed polygon be and Area of circumscribed polygon be An *.

In fig.1: n=3.    In fig.2: n=4.    fig.3: n=7. We can very well see that as n increases, the area of each polygon gets closer to the area of the circle.

As the number of sides increases, the areas of the inscribed and circumscribed polygons approach a fixed number, called a limit. This limit is equal to the area of the circle.

So in general we can write this as:

As n -> ∞; An -> Area of circle
As n -> ∞; An * -> Area of circle
lim   An = Area of circle = lim   An *
n->∞                               n->∞

From here on we will formally develop the theory of limits and look at ways to compute them.

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