James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

106 Chapter 2 Limits and Derivatives
ƒ
is in
here
y
5+∑
5
5-∑
Note that (1) can be rewritten as follows:
if 3 2 , x , 3 1 sx ± 3d then 5 2 « , f sxd , 5 1 «
and this is illustrated in Figure 1. By taking the values of x (± 3) to lie in the interval
s3 2 , 3 1 d we can make the values of f sxd lie in the interval s5 2 «, 5 1 «d.
Using (1) as a model, we give a precise definition of a limit.
FIGURE 1
0 3
x
3-∂ 3+∂
when x is in here
(x≠3)
2 Precise 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 sxd as x approaches a is L, and we write
lim f sxd − L
x l a
if for every number « . 0 there is a number . 0 such that
if 0 , | x 2 a | , then | f sxd 2 L | , «
Since | x 2 a | is the distance from x to a and | f sxd 2 L | is the distance from f sxd to
L, and since « can be arbitrarily small, the definition of a limit can be expressed in words
as follows:
lim x l a f sxd 5 L means that the distance between f sxd and L can be made arbitrarily small
by requiring that the distance from x to a be sufficiently small (but not 0).
Alternatively,
lim x l a f sxd 5 L means that the values of f sxd can be made as close as we please to L
by requiring x to be close enough to a (but not equal to a).
We can also reformulate Definition 2 in terms of intervals by observing that the
inequality | x 2 a | , is equivalent to 2 , x 2 a , , which in turn can be writ-
is true if and only if x 2 a ± 0, that is,
ten as a 2 , x , a 1 . Also 0 , | x 2 a |
x ± a. Similarly, the inequality | f sxd 2 L | , « is equivalent to the pair of inequalities
L 2 « , f sxd , L 1 «. Therefore, in terms of intervals, Definition 2 can be stated as
follows:
lim x l a f sxd 5 L means that for every « . 0 (no matter how small « is) we can find
. 0 such that if x lies in the open interval sa 2 , a 1 d and x ± a, then f sxd lies in
the open interval sL 2 «, L 1 «d.
We interpret this statement geometrically by representing a function by an arrow diagram
as in Figure 2, where f maps a subset of R onto another subset of R.
FIGURE 2
f
x a f(a) ƒ
The definition of limit says that if any small interval sL 2 «, L 1 «d is given around L,
then we can find an interval sa 2 , a 1 d around a such that f maps all the points in
sa 2 , a 1 d (except possibly a) into the interval sL 2 «, L 1 «d. (See Figure 3.)
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