2Limits and Continuity (1)

8/22/2019 2Limits and Continuity (1)

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Limits and Continuity

Definition

Evaluation of LimitsContinuity

Limits Involving Infinity


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Limit of the Function

Note: we can approach a limit from

left right both sides

Function may or may not exist at that point At a

right hand limit, no left

function not defined

At b left handed limit, no right

function defined

a b


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Can be observed on a graph.

Observing a Limit

View

Demo
http://localhost/var/www/apps/conversion/tmp/scratch_15/limits2.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits2.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits2.ggb


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Observing a Limit

Can be observed on a graph.


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Observing a Limit

Can be observed in a table

The limit is observed to be 64


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Non Existent Limits

Limits may not exist at a specific point for a

function

Set

Consider the function as it approaches

x = 0

Try the tables with start at0.03, dt = 0.01

What results do you note?

11( )

2y x

x


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Non Existent Limits

Note that f(x) does NOT get closer to a

particular value

it grows without bound

There is NO LIMIT

Try command on

calculator


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Non Existent Limits

f(x) grows without bound

View

Demo3


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Non Existent Limits

View

Demo 4


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Formal Definition of a Limit

The

For any (as close as

you want to get to L) There exists a (we can get as close as

necessary to c )

lim ( )x c

f x L

L

c


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Formal Definition of a Limit

For any (as close as you want to get to L)

There exists a (we can get as close as

necessary to c

Such that

( )f x L when x c


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Specified Epsilon, Required Delta


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Limit

We say that the limit of ( ) as approaches is and writef x x a L

lim ( )x a

f x L

if the values of ( ) approach as approaches .f x L x a

a

L

( )y f x


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c) Find2

3 if 2lim ( ) where ( )1 if 2x

x xf x f xx

-2

62 2

lim ( ) = lim 3x x

f x x

Note: f(-2) = 1

is not involved

23 lim

3( 2) 6

xx


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2

24( 4)a. lim

2xx

x

0

1, if 0b. lim ( ), where ( )

1, if 0x

xg x g x

x

20

1c. lim ( ), where f ( )

xf x x

x

0

1 1d. lim

x

x

x

Answer : 16

Answer : no limit

Answer : no limit

Answer : 1/2

3) Use your calculator to evaluate the limits


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The Definition of Limit-

lim ( )We say if and only ifx a

f x L

given a positive number , there exists a positive such that

if 0 | | , then | ( ) | .x a f x L

( )y f xa

LL

L

a a


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such that for all in ( , ),x a a a

then we can find a (small) interval ( , )a a

( ) is in ( , ).f x L L

This means that if we are given a

small interval ( , ) centered at ,L L L


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Examples

21. Show that lim(3 4) 10.

xx

Let 0 be given.

We need to find a 0 such that

if | - 2 | ,x then | (3 4) 10 | .x

But | (3 4) 10 | | 3 6 | 3 | 2 |x x x

if | 2 |3

x

So we choose .3

1

12. Show that lim 1.

x x

Let 0 be given. We need to find a 0 such that 1if | 1| , then | 1| .x

x

1 11But | 1| | | | 1| .x

xx x x

What do we do with the

x?


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1 31If we decide | 1| , then .2 22

x x

1And so


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The right-hand limit off(x), asx approaches a,equalsL

written:

if we can make the valuef(x) arbitrarily close

toLby takingx to be sufficiently close to the

right ofa.

lim ( )x a

f x L

a

L

( )y f x

One-Sided Limit

One-Sided Limits


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The left-hand limit off(x), asx approaches a,

equalsM

written:

if we can make the valuef(x) arbitrarily close

toLby takingx to be sufficiently close to theleft ofa.

lim ( )x a

f x M

a

M

( )y f x


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2

if 3( )2 if 3x xf x

x x

1. Given

3lim ( )

xf x

3 3lim ( ) lim 2 6

x xf x x

2

3 3lim ( ) lim 9

x xf x x

Find

Find3

lim ( )x

f x

Examples

Examples of One-Sided Limit


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1, if 02. Let ( )

1, if 0.

x xf x

x x

Find the limits:

0lim( 1)

xx

0 1 1

0a) lim ( )

xf x

0b) lim ( )

xf x 0

lim( 1)x

x

0 1 1

1

c) lim ( )x

f x 1

lim( 1)x

x

1 1 2

1d) lim ( )

xf x

1lim( 1)

xx

1 1 2

More Examples


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lim ( ) if and only if lim ( ) and lim ( ) .x a x a x a

f x L f x L f x L

For the function

1 1 1lim ( ) 2 because lim ( ) 2 and lim ( ) 2.x x x

f x f x f x

But

0 0 0lim ( ) does not exist because lim ( ) 1 and lim ( ) 1.x x xf x f x f x

1, if 0( )

1, if 0.

x xf x

x x

This theorem is used to show a limit does not

exist.

A Theorem


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Limit Theorems

If is any number, lim ( ) and lim ( ) , thenx a x a

c f x L g x M

a) lim ( ) ( )x a

f x g x L M

b) lim ( ) ( )x a

f x g x L M

c) lim ( ) ( )x a

f x g x L M

( )d) lim , ( 0)( )x a f x L Mg x M

e) lim ( )x a

c f x c L

f) lim ( )n n

x af x L

g) limx a

c c h) limx a

x a

i) lim n nx a

x a

j) lim ( ) , ( 0)x a

f x L L


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Examples Using Limit Rule

Ex. 23

lim 1

x

x

2

3 3

lim lim1

x x

x

2

3 3

2

lim lim1

3 1 10

x xx

Ex.1

2 1lim

3 5x

x

x

1

1

lim 2 1

lim 3 5

x

x

x

x

1 1

1 1

2lim lim1

3lim lim5

x x

x x

x

x

2 1 1

3 5 8


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More Examples

3 3

1. Suppose lim ( ) 4 and lim ( ) 2. Findx x

f x g x

3

a) lim ( ) ( )x

f x g x

3 3

lim ( ) lim ( )x x

f x g x

4 ( 2) 2

3

b) lim ( ) ( )x

f x g x

3 3

lim ( ) lim ( )x x

f x g x

4 ( 2) 6

3

2 ( ) ( )c) lim

( ) ( )x

f x g x

f x g x

3 3

3 3

lim 2 ( ) lim ( )

lim ( ) lim ( )

x x

x x

f x g x

f x g x

2 4 ( 2) 5

4 ( 2) 4


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Indeterminate forms occur when substitution in the limit

results in 0/0. In such cases either factor or rationalize the

expressions.

Ex.25

5lim

25x

x

x

Notice form00

5

5lim

5 5x

x

x x

Factor and cancel

common factors

51 1

lim5 10x x

Indeterminate Forms

M E l


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9

3a) lim

9x

x

x

9

( 3)

( 3)

( 3)= lim

( 9)x

x

x

x

x

9

9lim

( 9)( 3)x

x

x x

9

1 1lim

63x x

2

2 32

4b) lim

2x

x

x x

22

(2 )(2 )= lim

(2 )x

x x

x x

222= lim

x

x

x

2

2 ( 2) 41

( 2) 4

More Examples


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Co ri ht c 2003 Brooks/Cole a division of

Computing Limits

Ex.2

3 if 2lim ( ) where ( )1 if 2x

x xf x f xx

6

-2

2 2lim ( ) = lim 3

x xf x x

23 lim

3( 2) 6

xx

Note: f(-2) = 1

is not involved


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Computing LimitsEx.

Ex.

23

lim 1

x

x

2

3 3

lim lim1

x x

x

2

3 3

2

lim lim1

3 1 10

x xx

1

2 1lim

3 5x

x

x

1

1

lim 2 1

lim 3 5

x

x

x

x

1 1

1 1

2lim lim1

3lim lim5

x x

x x

x

x

2 1 1

3 5 8


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Indeterminate Forms:0

0

25

5lim

25x

x

x

Ex. Notice form

0

0

55

lim 5 5x

x

x x

51 1

lim5 10x x

Factor and cancelcommon factors


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Limits at Infinity

For all n > 0,1 1lim lim 0n nx xx x

provided that is defined.1

nx

Ex.2

2

3 5 1lim

2 4x

x x

x

2

2

5 13lim

2 4x

x x

x

3 0 0 3

0 4 4

Divide

by 2x

2

2

5 1lim 3 lim lim

2lim lim 4

x x x

x x

x x

x


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One-Sided Limit of a Function

The right-hand limit off(x), asx approaches a, equalsL

written:

if we can make the valuef(x) arbitrarily close toLbytakingx to be sufficiently close to the right ofa.

lim ( )x a

f x L

a

L

( )y f x


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One-Sided Limit of a Function

The left-hand limit off(x), asx approaches a, equalsM

written:

if we can make the valuef(x) arbitrarily close toLbytakingx to be sufficiently close to the left ofa.

lim ( )x a

f x M

a

M

( )y f x


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One-Sided Limit of a Function2

if 3( )2 if 3x xf x

x x

Ex. Given

3lim ( )

xf x

3 3lim ( ) lim 2 6

x xf x x

2

3 3lim ( ) lim 9

x xf x x

Find

Find

3

lim ( )x

f x


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Continuity of a Function

A functionfis continuousat the pointx = a if thefollowing are true:

) ( ) is definedi f a

) lim ( ) existsx aii f x

a

f(a)


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Continuity

A functionfis continuousat the pointx = a ifthe following are true:



a

f(a)


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A functionfis continuousat the pointx = a ifthe following are true:



) lim ( ) ( )x a

iii f x f a

a

f(a)

E l


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At which value(s) of x is the given function

discontinuous?

1. ( ) 2f x x

2

92. ( )3

xg xx

Continuous everywhereContinuous everywhere

except at3x

( 3) is undefinedg

lim( 2) 2x a

x a

and so lim ( ) ( )x a

f x f a

-4 -2 2 4

-2

2

4

6

-6 -4 -2 2 4

-10

-8

-6

-4

-2

2

4

Examples

1 if 0


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2, if 13. ( )

1, if 1

x xh x

x

1lim ( )x h x and

Thus h is not cont. atx=1.

1 1lim ( )x h x 3

h is continuous everywhere else

1, if 04. ( )

1, if 0

xF x

x

0

lim ( )x

F x

1and

0

lim ( )x

F x

1

ThusFis not cont. at 0.x

F is continuous everywhere else

-2 2 4

-3

-2

-1

1

2

3

4

5

-10 -5 5 10

-3

-2

-1

1

2

3


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Continuous Functions

Apolynomial functiony =P(x) is continuous at

every pointx.

A rational function is continuous

at every pointx in its domain.

( )( )

( )p x

R xq x

Iffandgare continuous atx = a, then

, , and ( ) 0 are continuous

at

ff g fg g a

g

x a


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Intermediate Value Theorem

Iffis a continuous function on a closed interval [a, b]andL is any number betweenf(a) andf(b), then there

is at least one numberc in [a, b] such thatf(c) =L.

( )y f x

a b

f(a)

f(b)

L

c

f(c) =

Example


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Example

2Given ( ) 3 2 5,

Show that ( ) 0 has a solution on 1,2 .

f x x x

f x

(1) 4 0(2) 3 0

ff

f(x) is continuous (polynomial) and sincef(1) < 0andf(2) > 0, by the Intermediate Value Theorem

there exists a c on [1, 2] such thatf(c) = 0.

Li it t I fi it


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Limits at Infinity

For all n > 0, 1 1lim lim 0n nx xx x

provided that is defined.1

nx

Ex.2

2

3 5 1lim

2 4x

x x

x

2

2

5 13lim

2 4x

x x

x

3 0 0 3

0 4 4

Divide

by2x

2

2

5 1lim 3 lim lim

2lim lim 4

x x x

x x

x x

x


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More Examples

3 2

3 2

2 3 21. lim

100 1x

x x

x x x

3 2

3 3 3

3 2

3 3 3 3

2 3 2

lim 100 1x

x x

x x x

x x x

x x x x

3

2 3

3 22

lim1 100 1

1x

x x

x x x

22

1

2


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0

2

3 2

4 5 212. lim

7 5 10 1x

x x

x x x

2

3 3 3

3 2

3 3 3 3

4 5 21

lim7 5 10 1x

x x

x x x

x x x

x x x x

2 3

2 3

4 5 21

lim5 10 1

7x

x x x

x x x

07

2 2 43. lim

12 31x

x x

x

2 2 4

lim12 31x

x x

x x xx

x x

42

lim31

12x

xx

x

2

12


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24. lim 1x

x x

2 22

1 1lim

1 1x

x x x x

x x

2 2

2

1lim

1xx x

x x

2

1lim

1x x x

1 10

I fi i Li i


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Infinite LimitsFor all n > 0,

1lim

nx a x a

1lim if is even

nx a

nx a

1lim if is odd

nx a

nx a

-8 -6 -4 -2 2

-20

-15

-10

-5

5

10

15

20

-2 2 4 6

-20

-10

10

20

30

40

More Graphs

-8 -6 -4 -2 2

-15

-10

-5

5

10

15

20
http://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.html


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Examples

Find the limits

2

20

3 2 11. lim

2x

x x

x

2

0

2 13= lim

2x

x x

3

2

3

2 12. lim2 6x

xx

3

2 1= lim

2( 3)x

x

x

-8 -6 -4 -2 2

-20

20

40


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Limit and Trig Functions

From the graph of trigs functions

( ) sin and ( ) cosf x x g x x

we conclude that they are continuous everywhere

-10 -5 5 10

-1

-0.5

0.5

1

-10 -5 5 10

-1

-0.5

0.5

1

limsin sin and limcos cosx c x c

x c x c


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Tangent and SecantTangent and secant are continuous everywhere in their

domain, which is the set of all real numbers

3 5 7, , , ,2 2 2 2

x

-6 -4 -2 2 4 6

-30

-20

-10

10

20

30

-6 -4 -2 2 4 6

-15

-10

-5

5

10

15

tany x

secy x


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Examples

2a) lim sec

x

x

2

b) lim secx

x

3 2c) lim tan

x

x

3 2

d) lim tanx

x

e) lim cotx

x

3 2g) lim cot

x

x

3 2

cos 0lim 0

sin 1x

x

x

4

f) lim tanx

x

1


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Limit and Exponential Functions

-6 -4 -2 2 4 6

-2

2

4

6

8

10

, 1xy a a

-6 -4 -2 2 4 6

-2

2

4

6

8

10

, 0 1xy a a

The above graph confirm that exponential

functions are continuous everywhere.

lim x cx c

a a


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Asymptotes

horizontal asymptotThe line is called a

of the curve ( ) if eihter

ey L

y f x

lim ( ) or lim ( ) .x xf x L f x L

vertical asymptoteThe line is called a

of the curve ( ) if eihter

x c

y f x

lim ( ) or lim ( ) .x c x c

f x f x


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Examples

Find the asymptotes of the graphs of the functions2

2

11. ( )

1

xf x

x

1(i) lim ( )

xf x

Therefore the line 1

is a vertical asymptote.

x

1.(iii) lim ( )x

f x

1(ii) lim ( )

xf x

.


is a vertical asymptote.

x


is a horizonatl asymptote.

y

-4 -2 2 4

-10

-7.5

-5

-2.5

2.5

5

7.5

10

1


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2

12. ( )

1

xf x

x

21 1

1(i) lim ( ) lim1x x

xf xx

1 1

1 1 1=lim lim .

( 1)( 1) 1 2x x

x

x x x


is a vertical asympNO t eT ot .

x

1(ii) lim ( ) .

xf x


is a vertical asymptote

x

(iii) lim ( ) 0.x f x Therefore the line 0

is a horizonatl asymptote.

y

-4 -2 2 4

-10

-7.5

-5

-2.5

2.5

5

7.5

10

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2Limits and Continuity (1)