33c Lims With Removable Disc

8/8/2019 33c Lims With Removable Disc

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Substitution Theorem E.1

If f is a polynomial function or a rational

function, then

provided that f(c) is defined. In the case of a

rational function, this means that the value ofthe denominator at c is not zero

)()(lim cfxfcx

!"


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#37

2

8lim

3

2

" x

x

x

)()(lim cfxftheoremonsubstitutiTrycx

!

"

0

0

22

82

2

8lim

33

2!

!

" x

x

x

..00

0undefinedisittoequalnotis


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You have an undefined rational

function?

SIMPLIFY IT!


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2 1 0 0 -8

2 4 8

1 2 4 0

#372

8lim

3

2

" x

x

x

What is x3 8 divided by x 2 equal to?

We are saying = x2 + 2x + 4 because (x2 + 2x + 4)(x-2) = x3 82

83

x

x

)2(

)42)(2(lim

2

8lim

2

2

3

2

!

"" x

xxx

x

x

xx

42lim 22

!"

xxx

12!4)2(222

!

Thus,


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#372

8lim

3

2

" x

x

x

)2(

)42)(2(lim

2

8lim

2

2

3

2

!

"" x

xxx

x

x

xx


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Holes Holes occur when there is a common factor on the

numerator and denominator. Step 2: REDUCE THE FUNCTION/SIMPLIFY IT, BUT

RECOGNIZE A HOLE. SET THE COMMON FACTOR TOZERO AND SOLVE FOR X TO FIND WHERE THE HOLE IS.

Example 1: 3)2(

)2(3

2

63)( !

!

!

x

x

x

xxf

)2(

)6(

)2)(2(

)6)(2()(

!

!

x

x

xx

xxxf

x 2 = 0 at x = 2 there is a hole. f(2) =3, so at (2,3) there is a hole

x + 2 = 0 at x = -2 there is a hole. f(-2) = -1, so at (-2,-1) there is a hole

2 61

2 2

!


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You have an undefined rational function

at a certain x value?

Simplify it

33lim)2(

)2(3lim

2

63lim)(lim

2222!!

!

!

"""" xxxx x

x

x

xxf


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Try 35, 36, 48-50; 57; 60


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Chapter 3.4

The study of continuity


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To say that a function f is continuous at x= c

means there is no interruption in the graph of

f at c.

In other words, at c, the graph in unbroken, so

there are no holes, asymptotes, jumps, or gaps


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Page 229

A function is continuous at c under the

following three conditions:

1. f(c) is defined

2. f(x) exists

3. f(x) = f(c)cx "

lim

cx "lim

Example: A function not continuous at c

( I I )a m c b

L


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Page 229

Continuity on an Open Interval

A function is continuous on an interval (a,b) if it iscontinuous at each point on the interval.

( I I )

a m c b

L


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Text book uses term open/closed

interval a lot

Open Interval: () brackets used to represent an open

interval. Does not include finite bounds

What is the largest number in the interval (3, 10)?

Closed Interval: [] brackets used to represent aclosed interval. Has finite bound

What is the largest number in the interval [3, 10]?


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One-sided limits Two types:

Left-Sided Limit: A limit as we approach c from the

left, so as x gets closer to c from smaller numbers

Notated:

Right-Sided Limit: A limit as we approach c from

the right, so as x gets closer to c from largernumbers

Notated:

Lxfcx

!

"

)(lim

Lxfcx

!

"

)(lim


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Determining Left and Right Handed

Limits

You can study it analytically using an x-y chart

You can graph it and study the limit visually

Intuitively


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u

!

2,3

2,1)(

2

xx

xxxf

x y

1.9 4.611.99 4.96

1.999 4.996

)(lim2

xfx "

Read: The limit of the

function as x approaches2 from the left

)(lim2

xfx "

Read: The limit of the

function as x approaches2 from the right

x y

2.1 0.92.01 0.99

2.001 0.999

= 5 = 1


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xxfGraph !)(

1

1

)(lim.12

xfx "

)(lim.22

xfx "

)(lim.32

xfx "

x y

1.9 1

1.99 1

1.999 1

=1

=2

= UNEx y

2.5 2

2.1 2

2.001 2


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Theorem 3.8 pg. 232

Let f be a function and let c and L bereal numbers.

if and only if

LxfandLxfcxcx

!!

""

)(lim)(lim

Lxfcx

!"

)(lim


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2)( ! xxfGraph

1

1

)(lim.12

xfx "

)(lim.22

xfx "

)(lim.32

xfx "

=1

=2

=UNE


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If the graph is defined on all values in an

interval, then it is continuous.

Can you find any points that are undefined on

the interval of study? On the whole thing?


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Page 229

A function is continuous at c under the

following three conditions:

1. f(c) is defined

2. f(x) exists

3. f(x) = f(c)cx "

lim

cx "lim


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Removable discontinuity- Holes

Occurs in rational functionsDo what you can to simplify a rational

function, so that you do no get

undefined for a function.

Pg. 227 #39; 49


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Examples

Page 230

Do a graph lik 3.22b


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3.4 Page 236

#s 2- 11; 15-17; 23-25

#s 27; 29; 31-36; 41;43; 51; 61; 62


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Regard page 240

Look at the definition of Infinite Limits and use

figure 3.34

Look for image to show here


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3.5 Infinite LimitsPg. 240

Infinite LimitsLet f be a function that is defined at every

real number in some open interval

containing c, except possibly c.

Then

means as you get nearer to c in x values,

the y-value increases.

g!"

)(lim xfcx


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3.5 Infinite LimitsPg. 240

Infinite LimitsLet f be a function that is defined at every

real number in some open interval

containing c, except possibly c.

Then

means as you get nearer to c in x values,

the y-value decreases.

g!"

)(lim xfcx


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Vertical asymptote pg. 242

A rational function is a quotient of two

polynomials, say

Then there is a vertical asymptote at x =c 00)( !{ cgbutcf

)(

)()(

xg

xfxh !


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Vertical Asymptote Definition

If f(x) approaches infinity (or negative infinity)

as x approaches c from the right or the left,

then the line at x=c is a VERTICAL ASYMPTOTE

Before, we said a vertical asymptote is a

vertical line that a function increases infinitely

but never crosses.


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Examples


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Homework 3.5

Page 1-3; 7;8; 15-19; 25-31(odd); 39; 59; 60

33c Lims With Removable Disc

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