Defining a Limit

 

Until now, we have been discussing limits in an intuitive way. We have been using words such as “very close to” and “closer and closer to” which are not so clear to meet the purpose of mathematics. Therefore, we need to look for an accurate definition of a limit. We begin that by the following discussion.

 

 

Suppose we have the function defined by:

 

We want to find the limit of this function as x approaches -3. Note that this function is the same as the function:

 

 

Now we can substitute in the second form of the function as long as x does not represent -3. Using the method we used before leads us to the conclusion that .

Substituting -3 for x in the first form of the function will give us 0/0, which is an indeterminate quantity. Thus, the function does not have a value at x = -3, and the value of the function is never -6. However, we can make the value of the function as close as we wish to -6 by giving x values close enough to -3. For example, if we want the value of the function to be inside , we will have to put x inside , so the value of the function will be different from -6 by less than 0.01. In the same way, we can make the value of the function closer to -6 by making x inside a smaller interval around -3, but not equal to -3.

 

 

That is the principal idea in the concept of limit, on which the next definition is based.

In short, if we have a positive number , no matter how small, we can find a positive number  such that when the difference between the value of x and 2 is less than , the difference between the value of the function and the value of its limit as x approaches 2 is less than , or in symbols:

.

That leads us to the following definition.

 

Definition

Let c be a point in an open interval , on which a function f is defined except possibly at c.

We say that the limit of the function f at c is L, if there is for every positive number  some positive number such that:

 

Remarks
  • The positive number is given first. Then we decide the other number . That means that the value of  is dependent on the value of , and not the contrary.
  • The definition included that , which does not require the function to be defined when x=a, and so the function may have a limit at a point although it is not defined at that point.
  • The definition said that the condition must be fulfilled for any positive number , not for only some values of . That is very important, and we will show the way of satisfying that condition in the following examples.
  • The previous definition gives us a method to prove that the limit of a function at some point has a specific value that we know before, but not to find the limit of the function at some point.
Let’s have some examples.

 

Example

Consider the function defined by . Prove that:

 

Solution

Example

If f is a function defined by:

Prove that

 

Solution

Example

Let f be the function defined by , where c is a constant. Prove that the limit of this function at any number a is equal to c.

 

Solution