88

CHAPTER 2
LIMITS AND DERIVATIVES
THE LIMIT OF A FUNCTION
Having seen in the preceding section how limits arise when we want to find the tangent to
a curve or the velocity of an object, we now turn our attention to limits in general and
numerical and graphical methods for computing them.
Let’s investigate the behavior of the function
defined by
for val
ues of
near 2. The following table gives values of
for values of
close to 2, but not
equal to 2.
From the table and the graph of
(a parabola) shown in Figure 1 we see that when
is
close to 2 (on either side of 2),
is close to 4. In fact, it appears that we can make the
values of
as close as we like to 4 by taking
sufficiently close to 2. We express this
by saying “the limit of the function
as
approaches 2 is equal to 4.”
The notation for this is
In general, we use the following notation.
DEFINITION
We write
and say
“the limit of
, as
approaches , equals ”
if we can make the values of
arbitrarily close to
(as close to
L
as we like)
by taking
x
to be sufficiently close to
(on either side of ) but not equal to .
Roughly speaking, this says that the values of
tend to get closer and closer to the
number
as
gets closer and closer to the number
(from either side of ) but
.
(A more precise definition will be given in Section 2.4.)
An alternative notation for
is
as
which is usually read “
approaches
as
approaches .”
a
x
L
f x
x
l
a
f x
l
L
lim
x
l
a
f x
L
x
a
a
a
x
L
f x
a
a
a
L
f x
L
a
x
f x
lim
x
l
a
f x
L
1
lim
x
l
2
x
2
x
2
4
x
f x
x
2
x
2
x
f x
f x
x
f
x
f x
x
f x
x
2
x
2
f
2.2
4
ƒ
approaches
4.
x
y
2
As
x
approaches 2,
y=
≈
x+2
0
FIGURE 1
x
3.0
8.000000
2.5
5.750000
2.2
4.640000
2.1
4.310000
2.05
4.152500
2.01
4.030100
2.005
4.015025
2.001
4.003001
f x
x
1.0
2.000000
1.5
2.750000
1.8
3.440000
1.9
3.710000
1.95
3.852500
1.99
3.970100
1.995
3.985025
1.999
3.997001
f x
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Notice the phrase “but
” in the definition of limit. This means that in finding the
limit of
as
approaches , we never consider
. In fact,
need not even be
defined when
. The only thing that matters is how
is defined
near
.
Figure 2 shows the graphs of three functions. Note that in part (c),
is not defined
and in part (b),
. But in each case, regardless of what happens at
, it is true that
.
EXAMPLE 1
Guess the value of
.
SOLUTION
Notice that the function
is not defined when
, but
that doesn’t matter because the definition of
says that we consider values of
that are close to
but not equal to .
The tables at the left give values of
(correct to six decimal places) for values of
that approach 1 (but are not equal to 1). On the basis of the values in the tables, we make
the guess that
M
Example 1 is illustrated by the graph of
in Figure 3. Now let’s change
slightly by
giving it the value 2 when
and calling the resulting function :
This new function
still has the same limit as
approaches 1 (see Figure 4).
0
1
0.5
x1
≈
1
y=
FIGURE 3
FIGURE 4
0
1
0.5
y=©
2
y
x
y
x
x
t
t
(
x
)
x
1
x
2
1
if
x
1
2
if
x
1
t
x
1
f
f
lim
x
l
1
x
1
x
2
1
0.5
x
f x
a
a
x
lim
x
l
a
f x
x
1
f x
x
1
x
2
1
lim
x
l
1
x
1
x
2
1
(c)
x
y
0
L
a
(b)
x
y
0
L
a
(a)
x
y
0
L
a
FIGURE 2
lim
ƒ=L
in all three cases
x
a
lim
x
l
a
f x
L
a
f a
L
f a
a
f
x
a
f x
x
a
a
x
f x
x
a
SECTION 2.2
THE LIMIT OF A FUNCTION

89
0.5
0.666667
0.9
0.526316
0.99
0.502513
0.999
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 Spring '09
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 Limits, lim, Limit of a function

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