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Limit of a function at a point.

published on February 16th, 2008 . by Vanaja

The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.

It is very important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus.

Definition

Let f(x) be a function of x. Let a and l be constants such that as x\rightarrow a, we have f\left(x \right)\rightarrow l. In such case we say that the limit of the function f(x) as x approaches a is l. we write this as

\lim_{x\rightarrow a}f\left(x \right)=l

In case , no such number l exist, then we say that \lim_{x\rightarrow a}f\left(x \right)does not exist finitely.

Illustration

Let a regular polygon of n sides be inscribed in a circle. The area of the polygon cannot be greater than the area of the circle., however large the number of sides of the polygon increases indefinitely the area of the polygon continually approaches the area of the circle. Thus the difference between the area of the circle and the polygon can be made as small as we please by sufficiently increasing the number of sides of the polygon.

We have \lim_{n\rightarrow \infty } (Area of the polygon of n sides)=Area of the circle.

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