2. Limits
P. K. Lamm
09/01/11 (13:57)
p. 1 / 8
Lecture Notes:
2. Limits
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of definitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
1. The Idea of a Limit
Definition (informal):
Let
f
be defined on an open interval about the point
x
0
, except possibly at
x
0
itself. If
f
(
x
) gets arbitrarily close to the number
L
for all
x
sufficiently close to
x
0
, we say
f
approaches
the limit
L
as
x
approaches
x
0
. We write this statement mathematically as
lim
x
→
x
0
f
(
x
) =
L,
or
f
(
x
)
→
L
as
x
→
x
0
,
or
x
→
x
0
⇒
f
(
x
)
→
L.
Note:
The limit of
f
(
x
) as
x
approaches
x
0
is specifically concerned with the behavior of
f
for
x
near
x
0
, and not at all concerned with the
value
of
f
at
x
0
.
Example 1.1:
Evaluate
lim
x
→
2
x
2

4
x

2
.
According to the definition of the limit, we need to determine what
y
value the curve
y
=
x
2

4
x

2
is
approaching as
x
approaches 2. To look more closely, we’ll make a table of some (
x, y
) values:
x
values
<
2
y
=
x
2

4
x

2
1.9
3.9
1.99
3.99
1.999
3.999
1.9999
3.9999
x
values
>
2
y
=
x
2

4
x

2
2.1
4.1
2.01
4.01
2.001
4.001
2.0001
4.0001
1
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2. Limits
P. K. Lamm
09/01/11 (13:57)
p. 2 / 8
As
x
values get closer and closer to the point 2, the
y
values appear to get closer and closer to 4. However,
since we didn’t check every single
x
value as it approached 2, we don’t know this for sure. Define
f
(
x
) =
x
2

4
x

2
,
and note that
f
(
x
) is not defined at
x
= 2. However, if
x
6
= 2, then
f
(
x
) =
x
2

4
x

2
=
(
x
+ 2)(
x

2)
x

2
= (
x
+ 2)
x

2
x

2
where
x

2
x

2
= 1
,
for all
x
6
= 2
.
(1)
Thus
f
(
x
) =
(
x
+ 2
,
x
6
= 2
undefined
,
x
= 2
,
and the function is exactly the same as the line
y
=
x
+ 2
except
at the point
x
= 2. We plot the line
y
=
x
+ 2 below on the left, and the graph of
y
=
f
(
x
) below on the right.
The definition of the limit specifically stated that we were not to be concerned with the value of the
function
f
at
x
0
(
x
0
= 2 in this case), only the value of
f
as
x
gets closer and closer to
x
0
= 2. We thus
have that
lim
x
→
2
x
2

4
x

2
= lim
x
→
2
(
x
+ 2)(
x

2)
x

2
= lim
x
→
2
(
x
+ 2)
·
1 = 4
,
where we have again used (1) since, in the limit, we never actually let
x
reach the point 2.
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 Calculus, Squeeze Theorem, Limits, Limit, lim, Limit of a function, lim g, P. K. Lamm

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