AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a 1

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

Documents

AIM: WHAT IS A LIMIT? HW#59 p. 218 – 219 # 14, 17, 18, 19, 20 p. 219 # 26 27, 29a 1