5.10 Limits

Limit is a concept which aims to determine nature of function at a point. This point is infinitesimally close to a declared or test point say “a”. We investigate nature of function when independent variable approaches the test value “a” and is not at “a”. In the nutshell, we seek to estimate value of function at x=a from a point which is very close to it. Definitely, neither “x” reaches “a” nor f(x) reaches a particular value, say, L. Thus, important thing is to understand that limit denotes correspondence of independent and dependent variables very near but not at the point of estimation.

We should keep in mind while studying limit that it is an estimation based on the behavior of function at points very near to the test point. Limit answers the question : “what would be function value at the test point from its behavior at a point which is very close?”. In answering this question, limit considers the nature of function as described by function rule and by estimating value at test point from either direction. This estimate or projection may, however, fail to match actual function value at the test point, if there is a jump or sudden change in function value i.e. when function is discontinuous at the test point. It does not matter. An estimate (limit) remains or exists – if it can be estimated – irrespective of whether it matches function value or not and whether there is a function value at all at the test point or not?

Delta – epsilon definition

Idea here is to express nature of function near a point, however, close. We can do this by choosing two very small positive numbers delta (δ) and epsilon (∈). We say that limit of function f(x) is L at x = a, if “x” approaches very close to “a”, then f(x) approaches very close to L. This means simultaneous closeness :

$$L-\delta <f\left(x\right)<L+\delta \text{for all x in}a-\in <x<a+\in$$

In modulus form :

$$|f\left(x\right)-L|<\delta \text{for all x in}|x-a|<\in$$

Limit of function is L, which may or may not be equal to value of function at x=a i.e. f(a). We shall discuss this aspect subsequently.

Notation

Limit of a function is denoted as :

$$\underset{x\to a}{\overset{}{\lim }}f\left(x\right)=L$$

We should read this notation carefully. It is the “limit of function” which is "equal to" L - not the function value. As far as function is concerned it is approaching “L” and value of function, f(a), may or may not be equal to “L”.

Nature of function

Nature of function is not known by its value at a point. Rather, it is known by the value it is likely to have at a neighboring point. Here, we consider hypothetical set up in order to understand the concept. Our job is to find the approaching value which the function will have and which can be represented as “L”. This we do by determining function value a little to the right towards test point if we approach the test point from left. Similarly, we approach the test point from right by determining function value a little to the left towards test point. If we approach the same value from either side from a very close point, then we say that limit of function at test point is “L”.

Important to note here is that this approaching value of function, L, at the test point, x=a, may or may not be the function value f(a). We should understand that the mechanism of piece-wise function definition allows us to define any function value for any point in the domain of function. Further, if test point is a singularity of domain (a point where function is not defined), then there is no function value at the test point.