Lec 3 =One Sided Limits Part 2 of Chapter 2

8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2

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ONE SIDED LIMITSChapter 2

Lecture 2 Part 2



INTRODUCTION

� So f ar in our discussion of the limit of a f unction

as the independent variable x approaches anumber a, we have been concerned with valuesof x as close to a and either greater than a orless than a, that is, values of x in an open

interval containing a but not at a itself

22



� Suppose, ho wever, that we have the f unction def ined by

� Because f( x) does not exist if x < 4, f is not def inedon any open interval containing 4. Thus, hasno meaning. If , ho wever, x is restricted to numbergreater than 4, the value of can be made asclose to 0 as we please by taking x suff iciently closeto 4 but greater than 4. In such a case we let x

approach 4 f rom the right and consider the righthand limit (or the one-sided limit f rom the right).

INTRODUCTION

4)( ! x x f

4lim4

p

x x

4 x

33





DEFINITION OF A LEFT-HAND LIMIT

� Let f be a f unction def ined at every number in

so

meo

pen interval (d ,a). Then the limitof

f( x),as x approaches a f rom the lef t, is L, written as

L x f a x

!

p

)(lim

55



NOTE

� W e ref er to as the tw o-sided limit to

distinguish itf r

om

one-sided limits.

� The limit theorems remain valid when ³ ³ isreplaced by either or

)(lim x f a xp

a xp

pa x

pa x

66



THEOREM

exists and is equal to L if and only if

and both exist and both are

equal to L.

)(lim x f a xp

)(lim x f a xp

)(lim x f a xp

77



Example:Determine if the limit exists.

Solution:

Since we conclude that

does not exist.

°̄

®

! x

x

xC 8.1

2

)(

x x x x

2lim)(lim1010

pp

!

20!

x x x x

8.1lim)(lim1010

pp

!

18!

)(lim)(lim1010

xC xC x x pp{ )(lim

10

xC xp

88



� Solution:

Since , then exists

and is equal to 3.


°̄

®

!

2

2

2

4

)( x

x

xh

2

11

4lim)(lim x xh x x

!

pp

2)1(4 !

3!

2

11

2lim)(lim x xh x x

!

pp

2)1(2 !

3!

)(lim)(lim11

xh xh x x pp

! )(lim1

xh xp

99



� Solution

Because , then exists andis equal to 0. Notice that g(0) = 2 which has no

eff ect on


°̄

®!

2)(

x

x g

)(lim)(lim00

x x g x x

!

pp

0!

x x g x x

pp

!

00

lim)(lim

0!

)(lim)(lim00

x g x g x x pp

! )(lim0

x g xp

)(lim0

x g xp

1010



Exercises: (MWF Classes)Determine if the limit exists.

±°

±̄

!

3

1

2

)()1 x f

°¯

!

t

t t f

4

4)()2

°¯®

! x

x x f 28

)()3

2

±°

±¯

!

r

r

r f

27

23

2

)()4

1111



°¯

!

2

2)()1 x f

°¯

!

s

s s f

3

3)()2

°®̄

! x

x x f 10

12)()3

±°

±¯

®

!2

2

11

03

)()4

t

t

t g

Exercises: (TTh Classes)Determine if the limit exists.

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Lec 3 =One Sided Limits Part 2 of Chapter 2

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