Friday September 16
−
Lecture 4
: Limits
(Refers to Section 2.1 to 2.3 in your text)
Students who have mastered the content of this lecture know about
:
An intuitive definition of limits of functions
,
onesided limits
,
infinite limits
,
limits at
infinity
,
vertical asymptotes
,
horizontal asymptotes
,
limit laws
,
the squeeze theorem
,
the
limit of sin(x)/x as x tends to 0
is 1
.
Students who have practiced the techniques presented in this lecture will be able to
:
Use
limit laws
and
use
elementary techniques to compute values of limits.
4.1
Definition
−
(
Intuitive definition
) Suppose we are given a function
f
(
x
) and a
number
a
. When we say that the “
limit of
f
(
x
)
as x approaches a is L
” written as
lim
x
→
a
f
(
x
) =
L
we mean that “
f
(
x
) becomes arbitrarily close to
L
as
x
approaches
a
from
either
side”.
(Note: There is a formal definition of the limit of a function in section 2.4 of your text. This is optional reading. We will not present
this in class nor is it required for Math 116 students to know this.)
4.1.1
Example – For the function
f
(
x
) whose curve is illustrated below see that the
limit of
f
(
x
) as
x
approaches 1 from either side is 6, while the limit of
f
(
x
) as
x
approaches
−
1 from either side is 3.
4.1.2
Examples : For each of the following we can determine the limits simply my
visualizing the curve of these functions.
a) lim
x
→
2
x
2
= 4
b) lim
x
→
a
x
=
a.
c) lim
x
→
0
(1/
x
) does not exist.
(That is, the function 1/
x
does not approach a number when
x
approaches 0.)
d) If
c
is a number and
f
(
x
) =
c
for all
x
, then lim
x
→
a
f
(
x
) =
c
for any
a
. Equivalently.
lim
x
→
a
c
=
c
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View Full Document4.2
Definition –
Onesided limits
. Say we are given a function
f
(
x
) and a number
a
. We
define the
lefthanded limit
as follows: It is the limit of the function when the value of
x
approaches
a
only from the
left
side. It is written as
lim
x
→
a
−
f
(
x
) =
L
We define the
righthanded limit
as follows: It is the limit of the function when the value
of
x
approaches
a
only from the
right
side.
The expression “limit of
f
(
x
) as
x
approaches
a
from the right
is
L
” is written as
lim
x
→
a
+
f
(
x
) =
L
Plotting the curve of a function often helps us determine lefthanded and righthanded
limits.
Note
: Sometimes there is no number
L
which is the limit of a function when
x
approaches a number
a
. When this is the case we say that the limit
does not exist
.
In the following graph of a curve
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 Spring '11
 RobertAndre
 Calculus, Limits, Limit, Limit of a function

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