Denition approach of learning new topics

11
Part II
Lecture 2
3 Examples of what is a Limit?
Instruction for this section : In the examples below, you might not understand
how few limits are used. Go through them without worrying if you are not
following something. Going further you will understand what we did in these
examples.
Let us consider an example to understand what Limits is exactly.
3.1 Polygon becomes circle 1
A circle of radius r is constructed. We can construct regular polygons inscribed
in that circle.
Now we can approximate the area of circle to be equal to area of a polygon
where number of sides is a very large number.
Suppose we have a polygon inscribed that has n sides.
Area of the polygon is
A = n · 1
2 r2 sin 2π n
The end to this process of continuously increasing the number of sides of the
inscribed polygon is same as trying to nd the limit (english word meaning) of
this process. The process limit or end would be that the polygon has become a
circle.
So as n → ∞, Area of polygon →Area of the circle.
This is written in notational form as lim Area of polygon = Area of Circle
n→∞
1
Area of Circle = lim
n→∞ 2 r2 n sin 2π n
r 2 sin 2π n
= lim
n→∞ 2
n
= r 2 sin 2π n
π lim
n→∞ 2π
n
= πr 2 · 1
[using
sin θ
lim = 1]
θ→∞ θ
1 You can view the animation of Polygon tending to a circle at TeachingMathematics