Module 2 : Limits and Continuity of  Functions
Lecture 4 : Limit at a point
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 Limit and Continuity of Functions
  Recall that, our aim is to understand a function by analyzing various properties of . For example, one would like to analyze:

   Does the 'graph' of have any 'breaks' ?

In this lecture we shall analyze the most important and fundamental concept: limit of a function, and shall see how it helps us to answer the above question.
   
4.1
Limit of a function concept :
 

Let us start with the following problem:

How to predict a suitable value of a function at a point, which may or may not be in its domain, by analyzing its values at points in the domain which are near the given point?

Let  . Let ,    may or may not be an element of . The question we want to answer is the following : Can we predict some 'suitable' value for at by looking at the values of at points close to in ? To answer this, let us assume that is defined at all points sufficiently near (may be not at ), for otherwise we have no data on the basis of which we can predict.
 


For example, this is true when A is an open interval or where is an open interval.

Next, we should clarify as to what do we mean by saying that a real number is a 'suitable value' for at ?

One way of interpreting this is to demand that the values comes closer to the number as the point comes 'closer' to . This immediately raises the following question: How do we interpret this mathematically ?  A natural way of doing this is to say that this closeness is achieved iteratively, i.e., we can come close to any point via sequences.
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