# Calculating Limits

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 |

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 -> ∞;

As n -> ∞;

OR

lim

n->∞ n->∞

*An*-> Area of circleAs n -> ∞;

*An **-> Area of circleOR

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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