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:
|| x 3
|| 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.
How close to 3 does x have to be so that f (x) differs from 5 by less than 0.1?
To obtain more detailed information about how f(x) varies when x is close to 3, we ask the following question:
| f (x) - 5 | < 0.1 if | x - 3 | < but x 3
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
| f (x) - 5 | < 0.1 if 0 < | x - 3 | <
| f (x) -5 | = | (2x - 1) - 5 | = | 2x - 6 | = 2| x - 3 | < 0.1
Notice that if 0 < | x - 3 | < (0.1)/2 = 0.05, then
| f (x) - 5 | < 0.1 if 0 < | x - 3 | < 0.05
| f (x) - 5 | < 0.01 if 0 < | x - 3 | < 0.005
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 | < 0.001 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
5 - < f (x) < 5 + whenever 3 - < x < 3 + (x 3)
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
Note that (1) can be rewritten as
and this is illustrated in Figure 1 (below).
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: