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