Let f be a real-valued function with f(x) = x + 5 for every real number x. Let’s consider the values of f(x) when the values of x are around 2, but x ≠ 2. ( See Table 1 below.)
Table 1
Values of f(x) around x = 2
Table 1 shows that as x approaches 2, the value of f(x) approaches to a certain value. Our intuition may estimate that f(x) approaches 7. In other words, if x approaches 2 then f(x) approaches 7. Also, we can say that when x is near but different from 2, f(x) is near 7. This relationship may be written as 
.
Note that in this case f(2) = 2 + 5 = 7 and 
. If so, how do they differ? What is the difference between, for instance, f(a) and 
? In the previous example , the difference may be slight. It may be more noticeable in the next example, when we consider 
. The values of g(x) around x = 2 (but x ≠ 2) are presented in the table below.
Table 2
Values of g(x) around x = 2
As Table 2 shows, when x approaches 2, g(x) approaches a certain value, that is 5. This relationship may be rewritten as 
. How about g(2)? If we substitute x = 2 into g(x) we get 
, whose value is not defined! In this example, the difference between 
and g(2) is more significant. As a matter of fact, the difference between the two is very substantial, considering that f(a) is the result of evaluating the value of f(x) exactly at x = 2 while evaluates the value of f(x). ) in the condition that x ≠ 2, although the value of x is “not far” from 2.
Existence of The Limit
In the second example, the value of g(2) is undefined. Stated another way, it has no value. Now, is it possible that the limit of a function has no value? This question is about the existence of the limit. As the next example will show, this may happen.
Let h be a real-valued function, defined for every real number x, with
The values of h(x) around x = 2 (but x ≠ 2) are presented in the table below.
Table 3
Values of h(x) around x = 2
Table 3 represents the following. If x is close to 2 but x < 2, the value of h(x) is close to, or around, 3. If x is close to 2 but x > 2, the value of h(x) is around -1. So, the value approached by h(x) depends on the direction from which 2 is approached. In this case, we say that 
does not exist. More specifically, we can state the following.
…………………… (1)
…………………… (2)
The forms (1) and (2) above are called one-sided limits. Furthermore, (1) and (2) are called the left-hand limit and right-hand limit, respectively.
In general, the limit of f(x) as x approaches a exists if and only if the value of the left-hand limit equals the value of the right-hand limit, that is:
there is 
