1. 1
Learning Objectives for Section 10.1
Introduction to Limits
The student will learn about:
■ Functions and graphs
■ Limits from a graphic approach
■ Limits from an algebraic approach
■ Limits of difference quotients.
2. 2
Functions and Graphs
A Brief Review
The graph of a function is the graph of the set of all ordered
pairs that satisfy the function. As an example, the following
graph and table represent the function f (x) = 2x – 1.
We will use
this point on
the next slide.
x f (x)
-2 -5
-1 -3
0 -1
1 1
2 ?
3 ?
3. 3
Analyzing a Limit
We can examine what occurs at a particular point by the
limit ideas presented in the previous chapter. Using the
function
f (x) = 2x – 1, let’s examine what happens near x = 2
through the following chart:
We see that as x approaches 2, f (x) approaches 3.
x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5
f (x) 2 2.8 2.98 2.998 ? 3.002 3.02 3.2 4
4. 4
Limits
In limit notation we have
.
3
1
2
lim
2
x
x
Definition: We write
L
x
f
c
x
)
(
lim
or
as x c, then f (x) L,
if the functional value of f (x) is close to the single real
number L whenever x is close to, but not equal to, c (on
either side of c).
2
3
5. 5
One-Sided Limits
■ We write
and call K the limit from the left (or left-hand limit) if
f (x) is close to K whenever x is close to c, but to the left
of c on the real number line.
■ We write
and call L the limit from the right (or right-hand limit)
if f (x) is close to L whenever x is close to c, but to the
right of c on the real number line.
■ In order for a limit to exist, the limit from the left and
the limit from the right must exist and be equal.
K
x
f
c
x
)
(
lim
L
x
f
c
x
)
(
lim
6. 6
Example 1
4
)
(
lim
2
x
f
x
2
)
(
lim
2
x
f
x
Since these two are not the
same, the limit does not exist
at 2.
4
)
(
lim
4
x
f
x
4
)
(
lim
4
x
f
x
Since the limit from the left and
the limit from the right both
exist and are equal, the limit
exists at 4:
2 4
2
4
On the other hand:
4
)
(
lim
4
x
f
x
8. 8
Limit Properties
Let f and g be two functions, and assume that the following
two limits exist and are finite:
Then
the limit of the sum of the functions is equal to the sum of
the limits.
the limit of the difference of the functions is equal to the
difference of the limits.
M
x
g
L
x
f
c
x
c
x
)
(
lim
and
)
(
lim
9. 9
Limit Properties
(continued)
the limit of a constant times a function is equal to the
constant times the limit of the function.
the limit of the product of the functions is the product of
the limits of the functions.
the limit of the quotient of the functions is the quotient
of the limits of the functions, provided M 0.
the limit of the nth
root of a function is the nth
root of the
limit of that function.
10. 10
Examples 2, 3
13
8
1
3
lim
2
lim
1
3
2
lim
4
4
4
x
x
x x
x
x
x
2 2
2 2 2
lim 3 lim lim3 4 6 2
x x x
x x x x
From these examples we conclude that
1.lim ( ) ( )
2.lim ( ) ( )
x c
x c
f x f c
r x r c
f any polynomial function
r any rational function with a
nonzero denominator at x = c
11. 11
If and , then
is said to be indeterminate.
The term “indeterminate” is used because the limit may or
may not exist.
Indeterminate Forms
0
)
(
lim
x
f
c
x
)
(
)
(
lim
x
g
x
f
c
x
0
)
(
lim
x
g
c
x
It is important to note that there are restrictions on some of
the limit properties. In particular if 0
)
(
lim
x
r
c
x
then finding may present difficulties, since the
denominator is 0.
( )
lim
( )
x c
f x
r x
12. 12
Example 4
This example illustrates some techniques that can be useful for
indeterminate forms.
2
2 2 2
4 ( 2)( 2)
lim lim lim( 2) 4
2 2
x x x
x x x
x
x x
Algebraic simplification is often useful when the numerator and
denominator are both approaching 0.
13. 13
Let f (x) = 3x - 1. Find
Difference Quotients
.
)
(
)
(
lim
0 h
a
f
h
a
f
h
14. 14
Let f (x) = 3x - 1. Find
Solution:
Difference Quotients
.
)
(
)
(
lim
0 h
a
f
h
a
f
h
1
3
3
1
)
(
3
)
(
h
a
h
a
h
a
f
1
3
)
(
a
a
f
h
a
f
h
a
f 3
)
(
)
(
.
3
3
lim
)
(
)
(
lim
0
0
h
h
h
a
f
h
a
f
h
h
15. 15
Summary
■ We started by using a table to investigate the idea of a
limit. This was an intuitive way to approach limits.
■ We saw that if the left and right limits at a point were the
same, we had a limit at that point.
■ We saw that we could add, subtract, multiply, and divide
limits.
■ We now have some very powerful tools for dealing with
limits and can go on to our study of calculus.
