Limit:A Bud for Calculus

Calculus is one of the most fascinating, rigorous and importance subject in Mathematics. The two independent founders of calculus Sir Issac Newton and Gottfried Leibniz have done the great contribution for mathematics and the whole world by founding this amusing subject. Newtonian mechanics stands on the base of calculus and goes along with calculus. The pillar on which calculus stands is the concept of limit. The limit concept not only is the base for calculus but also the savior point when other subjects in calculus get puzzled. Limit is the base and ultimate savior. I say limit the mother of calculus.

There are many things in Mathematics that can't be represented in concrete form or can not be perceived at some glances. To provide the meaning and intuition to good degree of approximation, limits are used. What will happen if you built a tower of millions of millions of meters of height? I don't know what other things may happen but I am quite sure you will take help of limits to know what happens. Limit is approximation and the approximation tends to exact reality.

The mathematical definition of limit is mathematically exquisite. But we don't have math sense organs; so how beautiful the definition of limits may be we need some kind of intuition and illustration behind it.

 Let's understand limit a bit more practically. Limit is the result that we get if we go near and near to the situation or condition.

Let me draw a circle and make a inscribed regular polygon of n sides as shown in figure.

Here, area of the regular polygon < area of circle

How will it be if the polygon has  infinitely many sides?It seems puzzled at first hearing but ask limit to help and you will get clear view of the situation. In addition you will too learn what limit actually means.If the regular polygon has three sides it seems inscribed triangle. If it has four sides it seems inscribed square and so on.If we join every vertices of n-sided polygon to center of circle we get 'n' number of triangles. And the sum of all the 'n' triangles give total area of the polygon.

What happens if we make n infinitely large? Does the area goes to infinity? Or does the area vanishes? Lets put n = infinity and see what happens!

Oh dear! What is it? Looking at the first term I feel that I get huge enormous area. But second term says the area vanishes. It is like infinity*0. I don't know what result it gives. This is really confusing expression we have got. And using simple algebra it takes us nowhere. But don't worry. We have this awesome tool called limit and we make use of it here.

Yes the task is solved. The infinity sided polygon tends to be a circle as we get 'A' as area of the circle. Pretty interesting, isn't it?This is the limit concept and how it is beneficial to all we crazy explorers. This is its vigor.

In the next article I use limit in world's population case. How will the population and world go after thousands of years? Will all the available resources vanish? Will the mankind end due to lack of resources? We will find out in next article of limit.

Thank you.