1
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
realUse the rational zero test and synthetic division to find the zeros of
each polynomial function
31. ( )f x x x
32. ( ) 2 3f x x x
4 3 23. ( ) 4 9 16 20f x x x x x
2
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
1f x
x
1. ( 0)GRAPH It never crosses the y axis vertical asymptoteat x
FUNCTION division by zerocreates the vertical asymptote
2. * 0 , .
* 0 , .
,
GRAPH As x approaches from the left the function value f x decreases without bound
As x approaches from the right the function value f x increases without bound
FUNCTION Thebigger the x value the smaller the y value
.
, .
3 Because the function decreases without bound from the left and increa
th
ses w
e fun
ithout bo
ction is n
und
from t ot continhe righ uoust
0?How does the function behave as x approaches
' : limx c
What s to come f x
:A few observations on the behavior of the function and its graph
limit 0.
, , limit
.
The does not exist if f x is not approaching a unique real number as x approaches
In general if a function is undefined at a certain x value the does not exist as x approaches
that x value
0 0
1 1lim lim limit .x x
and the does existx x
3
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
A limit gives us language for describing how function values (y-values) behave as the domain values (x-values) approach some particular number (or ∞ or -∞).
In other words, a limit tells us:
• what the y-values of a function are approaching as x approaches some value c (but not necessarily equal to c) or it tells us
• the intended y-value of a function as x approaches c (we will see this when x approaches a hole).
: limx c
Notation f x L
2limit : lim 3 2
xEstimatethe numerically x
:
3 2 :
2 2 .
PROCEDURE
Let f x x and construct a table that shows the values of f x for two sets of values
one set that approaches from the left and one that approaches from the right
1.9 1.99 1.999 2.0 2.001 2.01 2.1
3.700 3.970 3.997 ? 4.003 4.030 4.300
x
f x
limi
2 , 4
, t 4.
.It appears that the closer x gets to from both sides the c
estimate t
loser f x gets to
So y he to an o beu c
"What is the function value of
3 2 as x approaches 2"f x x
4
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
1
limx
f x
2
2
3
2
limx
f x x
f x
This function is a parabola whose domain is all real numbers.
e know that there are no assymptotes or breaks in this curve -
it is a continuous function (there is nothing in this equation that can
ca
W
use a zero in the denominator). Therefore, the limit as x approaches
any real number (in this case -3) will be a difinite function value
and thus we can use subsitution to evaluate the limit:
2
33 2 11lim
xf x
The limit as x approaches a number whose function value is definite:
1.
2.
5
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
1
limx
f x
1
limx
f x
1
1
lim
limx
x
f x
f x
When asked to find a limit as x approaches an asymptote,
we say the limit does not exist, or give or - as an answer,
accordingly. *** Be sure to check the limit coming from the
left and from the r
ight.
The limit as x approaches an asymptote: (limits that fail to exist)
1.
2.
1
limx
f x
2 1 1 31.5
1 1 2
3
limx
f x
2
2 2.5
43 1
1
The limit does
not exist as x
approaches
1
The limit does not exist
as x approaches
limx
f x
2
limx
f x
0
6
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
Another example of when a limit fails to exist:
0
limx
xf x
xf x
No matter how close x gets to 0, there will be both positive and negative
x-values that yield 1 1.
This implies that the limit does not exist.
f x and f x
: Parent signature and test corrections on loose-leaf!!!
However, we can call this type of behavior a one-sided limit:
0
0
lim 1
lim 1x
x
f x
f x
2
limit 1.
4 , 1
4 , 1
Find the of f x as x approaches
x xf x
x x x
:
you are concerned with the value of
1 rather than 1.
REMEMBER
f x
near x at x
1 1
2
1 1
1
2
Using :
lim lim 4 4 1
lim li
3
3
lim 3
m 4 4 1 1
x
x x
x x
direct substitution
f x x
f x x x
f x
7
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
The limit as x approaches a hole: :
0A hole is the result of
0#
An asymptote is the result of 0
RECALL
2
2
2
6
x xf x
x x
1.
2.
: 2Hole x
3
3
lim
limx
x
f x
f x
3limx
f x
2
limx
f x
.6
1
2 3
2x
x
x
x
3 2
1
1
1limx
x x xf x
xf x
2
1 2 1 3.6
2 33 5x
xor
The actual value of is immaterial.
Therefore, if c is a hole the limit still exists;
it is the closest function value to the hole.
f x
: 3Vertical assymtote x
8
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
Some more examples:
2, 3
0, 3
xf x
x
The actual value of is immaterial. Therefore, if c is a hole, the limit still exists;
it is the closest function value to the hole and is written as follows:
limx c
f x
f c L NOT as f c L
2
3limx
f x
23f
This is the reason for limits. We don't necessarily want the actual value of the function at
the given x-value, but we want to know what value the function is approaching or how the
function is behaving as it approaches that value.x
*Because 2 for all other than 3 and because the
value of 3 is immaterial, it follows that the limit is 2.
*The fact that 3 0 has no bearing on the existence or value
of the limit as approache
f x x x
f
f
x
s 3.
1
limx
f x
2
9
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
4
1f x
x
0
limx
f x
limx
f x
1
limx
f x
10 4
1Think of it in terms of the equation: as x gets very big, what happens to ?
xIt gets very small, closer and closer to 0, but never negative.
2
1
3f x
x
3
limx
f x
limx
f x
3
limx
f x
What is the domain of this function?
: , 3 , 3,D
limx
f x
00
4
limx
f x
1
limx
f x
11
4
10
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
SUMMARY OF THE RELATIONSHIP BETWEEN HORIZONTAL AND VERTICAL ASYMPTOTES AND LIMITS:
HORIZONTAL ASYMPTOTE
VERTICAL ASYMPTOTE
y b f x
x a f x
y b
x a
lim
limx
x
f x
f x
lim
limx a
x a
f x
f x
bb
11
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
END-BEHAVIOR MODEL (EBM) FOR RATIONAL FUNCTIONS:
Sometimes we are not concerned with the behavior of near a specific point,
but rather with how the values of behave as increases or decreases without bound.
This is called the END BEHAVIOR of th
f x
f x x
e function because it describes how the function
behaves for values of that are far from the origin. x
HOW DO WE USE THIS END-BEHAVIOR MODEL (EBM):
We write another function that behaves just like the original function at the ends, by
taking the term with the highest degree in the numerator over the term with the highest degree
in the denominator and simplifying; this imitates the graph at the "ends", not in the middle;
( )
( )
as x the right end
as x the left end
Find an End-Behavior Model function for and call it :g x
EXAMPLE OF (EBM):
5
2322
33
xg x
xx
5 4 2
2
2 1
3 5 7
x x xf x
x x
32
3g x x
: The functions will
not always look the same.
NOTE
12
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
WHEN HAVE WE USED THE “EBM” IN A SIMILAR FASHION?
To find horizontal asymptotes.
5 4 2
2
2 1
3 5 7
x x xf x
x x
32
3g x x
LET’S EXAMINE THE PREVIOUS EXAMPLE MORE CLOSELY:
Find all asymptotes of the original function :f x:
:
denominator 0
RECALL
To find Vertical Assymtotes
Set and solve for x values
:
. , 0
. ,
. ,
n
m
To find Horizontal Assymtotes
a If n m y
ab If n m y
b
c If n m no horizontal assymtotes
2Denominator 3 5 7 0
0
x x
There areno x values such that theden
5, 2n m
n m
There are no horizontal asympototes
13
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
2
2
4 4
33
tg t
t
2
2
4 3 1Use the EBM method to find lim when
3 2 5t
t tf t f t
t t
As t , a problem may only occur if there exits a horizontal asymptote,
,as t , the function would also approach infinity and therefore
the limit does not exist. So, to find the limit of a
otherwise
function as t , we will
look for any horizontal asymptotes using the EBM:
4There exists a horizontal asymptote at
3y
4lim
3tf x
Let's look at the graph of to verify our answer:f t
4
3y
As t is approaching infinity, the function
4values are getting very close to
3
14
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
Find the limit (if it exists). If the limit does not exist. explain why:
4
2
2
2
2
2
510. lim
25 2
13. lim6 15 3
14. lim4
418. lim
3
5 6 320. lim
2 4
9 1022. lim
2 4 3
x
x
x
x
x
t
xx
xx
x
x
x
x x
x x
t t
t t
05
6
3
3
2
1
3
limit .The does not exist
15
AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a
Review for Test:
11
2f x
x
2
2
1
lim
lim
lim
lim
lim
x
x
x
x
x
Domain
f x
f x
f x
f x
f x
Use the EBM to find the limit (if it exists).
If the limit does not exist. explain why:
2
2
2 6lim
1x
x
x
2
* limits
3 2
2 1
lim
limx
x
Find the following of
xf x
x
f x
f x
This function approaches
different horizontal asymptotes
to the left and to the right.
2
3 3 3:
2 22:
3 3
2 2
x xEBM g x
xxHorizontal asymptotes
y and y
3
2y
3
2y 3
2
3
2
, 2 , 2,
11
2
3
To find horizontal asymptotes in this example,
1you must combine 1 as follows:
22 1 21 1 1 2 1
1
2 1 2 2
: 11
2 2
horizontal asymtote y
xx x x x
x x x x x
To find the limit as ,
recall that you must find any
horizontal asymptotes.
x
2
2
2 6lim
2 1x
x
x x
2
2
2: 2
. . 2
xEBM g x
xhoriz asymp y
lim 2x
f x