Lec 3 =One Sided Limits Part 2 of Chapter 2
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Transcript of Lec 3 =One Sided Limits Part 2 of Chapter 2
8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
http://slidepdf.com/reader/full/lec-3-one-sided-limits-part-2-of-chapter-2 1/12
ONE SIDED LIMITSChapter 2
Lecture 2 Part 2
8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
http://slidepdf.com/reader/full/lec-3-one-sided-limits-part-2-of-chapter-2 2/12
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
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8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
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� 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
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8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
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8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
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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
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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
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8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
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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
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8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
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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
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8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
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� Solution:
Since , then exists
and is equal to 3.
Example:Determine if the limit exists.
°̄
®
!
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
8/6/2019 Lec 3 =One Sided Limits Part 2 of Chapter 2
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� Solution
Because , then exists andis equal to 0. Notice that g(0) = 2 which has no
eff ect on
Example:Determine if the limit exists.
°̄
®!
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
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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