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:

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

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

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

that is,

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:

Similarly,

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

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

Note that (1) can be rewritten as

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 - .

THE FORMAL DEFINITION OF A LIMIT:

Using (1) as a model, we give a precise (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
whenever

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if for every number there is a corresponding number such that

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