Calculus In Brief:
Of all the concepts learned in Calculus, the one that gives way to many more ideas is that of the limit. But what is a limit? What does it mean when someone says, "the limit of 5x as x approaches 2 is 10"? The truth lies within...
THE FORMAL DEFINITION OF A LIMIT
Note that some like to refer to this concept as "The Precise Definition of a Limit"
Let's begin our journey into this monumental topic by considering the following function:

f(x) = 2x-1 if x 3
6 if x = 3

Intuitively it is clear that when x is close to 3 but x 3, then f(x) is close to 5, and so lim x --> 3 f(x) = 5.
To obtain more detailed information about how f(x) varies when x is close to 3, we ask the following question:

How close to 3 does x have to be so that f (x) differs from 5 by less than 0.1?

The distance from x to 3 is | x - 3 | and the distance from f (x) to 5 is | f (x) -5 |, so our problem is to find a number such that

| f (x) - 5 | < 0.1      if      | x - 3 | <      but x 3

If | x - 3 | > 0, then x 3, so an equivalent formulation of our problem is to find a number such that

| f (x) - 5 | < 0.1     if     0 < | x - 3 | <

Notice that if 0 < | x - 3 | < (0.1)/2 = 0.05, then

| f (x) -5 | = | (2x - 1) - 5 | = | 2x - 6 | = 2| x - 3 | < 0.1

that is,
| f (x) - 5 | < 0.1     if     0 < | x - 3 | < 0.05

Thus, an answer to the problem is given by = 0.05; that is, if x is within a distance of 0.05 from 3, then f (x) will be within a distance of 0.1 from 5.
If we change the number 0.1 in our problem to the smaller number 0.01, then by using the same method we find that f (x) will differ from 5 by less than 0.01 provided that x differs from 3 by less than (0.01)/2 = 0.0005:

| f (x) - 5 | < 0.01     if     0 < | x - 3 | < 0.005

Similarly,
| f (x) - 5 | < 0.001     if     0 < | x - 3 | < 0.0005

If, instead of tolerating an error of 0.1 or 0.001, we want accuracy to within a tolerance of an arbitrary positive number (the Greek letter epsilon), then we find as before that

| f (x) - 5 | <     if     0 < | x - 3 | < = /2

This is a precise (or formal) way of saying that f (x) is close to 5 when x is close to 3 because (1) it says that we can make the values of f (x) within an artibitrary distance from 5 by taking the values of x within a distance /2 from 3 (but x 3).
Note that (1) can be rewritten as
5 - < f (x) < 5 +     whenever     3 - < x < 3 +     (x 3)

and this is illustrated in Figure 1 (below).

Figure 1
By taking the values of x (3) to like in the interval (3 - , 3 + ) we can make the values of f (x) lie in the interval (5 - .

Using (1) as a model, we give a precise (formal) definition of a limit,


THE FORMAL DEFINITION OF A LIMIT:
Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f (x) as x approaches a is L, and we write
if for every number there is a corresponding number such that
   whenever   




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