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O
1) Rxo
1.1) Rxo
3
32)(
2
x
xxxf
),3()3,( A .
1)(3
)3)(1()(
xxf
x
xxxf
1 f .
f ),4()4,( B .
1
x A .
x 3,
3, 3 . ,
)(xf 4, 4
, x .
, 3 x ,
4 )(xf .
)(xf , x 3 4
:
4)(lim3
xfx
f ),(),( oo xx
fC ( 2).
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2
ox x , l
)(xf .
)(xf , x ox l
)(xf ox l :
lxfoxx
)(lim
:
i) ox f ( 3)
ii) )(lim xfoxx
f
: ),(),( oo xxa ),( oxa ),( ox .
iii) ox f , )( oxf .
)(lim xfoxx
)( oxf . ( 4
5).
3
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4
5
3,
3,
5
3)(
2
x
x
x
xxf . 6
f .
x 3 ( x
3, )3,(x ) )(xf 6.
x 3 ( x 3,
),3( x ) )(xf 2.
)(xf , x 3
6 6)(lim3
xfx
.
)(xf , x 3
2 2)(lim3
xfx
.
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6
f
),(),( oo xxa .
i) )(xf 1l x
ox , ),( oxa , )(xf , x ox
1l 1)(lim lxfoxx
.
ii) )(xf 2l x
ox , ),( oxx , )(xf , x ox
2l 2)(lim lxfoxx
.
( 7)
y
x
O
y'
x
x
f(x) 1 3 5
2
-3
6
f(x)
x
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7
To 1)(lim lxfoxx
f ox ,
2)(lim lxfoxx
f ox .
1l 2l f ox .
:
lxfxflxfooo xxxxxx
)(lim)(lim)(lim
y
x
O
x
f(x)
xo x
l2 f(x)
x
l1
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1.2*) f Rxo
.
f ),(),( oo xxa .
f ox Rl
( )
),(),( oo xxax ),(),( oooo xxxxx :
),()( llxf
( 8)
8
),(),( oooo xxxxx
),( oo xxx oxx oxx0
),()( llxf lxf )(
:
f ),(),( oo xxa .
y
x
O
x
l-
xo
l f(x)
x xo- xo+
l+
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f ox Rl lxfoxx
)(lim ,
0 0 ),(),( oo xxax
oxx0 lxf )( .
* f ox .
f ox , , .
:
i) 0)(lim)(lim
lxflxfoo xxxx
ii) lhxflxf ohxx o
)(lim)(lim0
:
f ),(),( oo xxa .
:
lxfxflxfooo xxxxxx
)(lim)(lim)(lim
, )(lim)(lim xfxfoo xxxx
, ox f .
f ),( oxa .
:
)(lim)(lim xfxfoo xxxx
f ),( ox .
:
)(lim)(lim xfxfoo xxxx
, :
f ox . :
i) f
),(),( oo xxa .
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ii) f ),( oxa
. ( f
),( ox ).
iii) f ),( ox
. ( f
),( oxa ).
xxf )( ( ) cxg )( , Rc
( ) R . :
oxx
xxfo
)(lim oxx
xxo
lim
cxgoxx
)(lim ccoxx
lim
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1) f ),(),( oo xxa
3)(lim 3
xfoxx
23)(lim xfoxx
.
R )(lim xfoxx
.
:
)(lim xfoxx
:
113010103
0)1)(1)(3(0)1)(3(0)3()3(
03333)(lim)(lim
22
2323
xfxfoo xxxx
2) x
xxxf 2
2
1)( .
) fC .
) fC : )(lim0
xfx
)(lim0
xfx
.
) , , )(lim0
xfx
.
:
x
xxxf 2
2
1)( , *Rx :
0,
0,
12
1
12
1
)(2
2
x
x
x
xxf
) f .
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10 : - http://sciencephysics4all.weebly.com/
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) :
1)(lim0
xfx
1)(lim0
xfx
) )(lim)(lim00
xfxfxx
, f , x 0.
3) 3
96)2()(
2
x
xxxxf .
) f .
) , , )(lim3
xfx
( fC ).
:
3
96)2()(
2
x
xxxxf , ),3()3,( x
:
y
x
O
y'
x
-1
2
1
3
-2
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3,
3,
2
2)(
),3()3,(,3
3)2()(
),3()3,(,3
)3()2()(
2
x
x
x
xxf
xx
xxxf
xx
xxxf
) f .
) :
1)(lim3
xfx
1)(lim3
xfx
)(lim)(lim33
xfxfxx
, f , x 3.
y
x
O
y'
x
-1
2
1
2
3
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1.3)
():
1)
i. 0)(lim
xfoxx
, 0)( xf ox . ( 1)
ii. 0)(lim
xfoxx
, 0)( xf ox . ( 1)
1
y
x
O
x
xo
l
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1
2) f g ox
)()( xgxf ox . )(lim)(lim xgxfoo xxxx
( 2 2).
2
y
x O x xo
l
y
y
x
O
x
xo
y
Cf
Cg
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2
(1) (2)
():
f g ox .
)
i. 0)( xf ox , 0)(lim
xfoxx
.
ii. 0)( xf ox , 0)(lim
xfoxx
.
)
i. )(lim)(lim xgxfoo xxxx
, )()( xgxf ox .
ii. )(lim)(lim xgxfoo xxxx
, )()( xgxf ox .
) )()( xgxf ox , )(lim)(lim xgxfoo xxxx
.
3) f g : lxfoxx
)(lim
mxgoxx
)(lim . :
y
x
O
x
xo
y
Cf
Cg
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15 : - http://sciencephysics4all.weebly.com/
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i. mlxgxfxgxfooo xxxxxx
)(lim)(lim)()(lim
ii. lcxfcxcfoo xxxx
)(lim)(lim , Rc
iii. mlxgxfxgxfooo xxxxxx
)(lim)(lim)()(lim
iv. 0)(lim
xgoxx
m
l
xg
xf
xg
xf
o
o
o
xx
xx
xx
)(lim
)(lim
)(
)(lim
v. lxfxfoo xxxx
)(lim)(lim
vi. kkxx
k
xxlxfxf
oo
)(lim)(lim ( 0)( xf
ox ).
vii. vv
xx
v
xxlxfxf
oo
)(lim)(lim , *Nv
: vov
xxxx
o
lim .
4) : ov
v
v
v axaxaxaxP
1
1
1 ...)( , 0va
ox . : )()(lim oxx
xPxPo
:
:
)(...
lim)(lim...)(lim)(lim
lim)(lim...)(lim)(lim
)...(lim)(lim
1
1
1
1
1
1
1
1
1
1
1
1
ooo
v
ov
v
ov
oxxxx
v
xxv
v
xxv
oxxxx
v
vxx
v
vxx
o
v
v
v
vxxxx
xPaxaxaxa
axaxaxa
axaxaxa
axaxaxaxP
oooo
oooo
oo
5) )(xP )(xQ x Rxo 0)( oxQ ,
:
)(
)(
)(
)(lim
o
o
xx xQ
xP
xQ
xP
o
.
:
:
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16 : - http://sciencephysics4all.weebly.com/
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)(
)(
)(lim
)(lim
)(
)(lim
o
o
xx
xx
xx xQ
xP
xQ
xP
xQ
xP
o
o
o
1) :
i) 234
23
1 45
34lim
xxx
xxx
x
, ii)
82
83lim
2
2
2
xx
xx
x
:
i) 234
23
45
34)(
xxx
xxxxf
1x
.
(3iv) 1ox . :
xx
x
xx
x
xxx
xxx
xxx
xxx
xxx
xxxxf
4
3
)4(
3
)4)(1(
)3)(1(
)45(
)34(
45
34)(
2222
2
234
23
3
2
3
2
141
31
4
3lim)(lim
2211
xx
xxf
xx.
ii) 2x :
82
83)(
2
2
xx
xxxf .
(3iv) 2ox . :
)83(
)2(
)4(
2
)83)(4(
)2(2
)83)(4)(2(
)2)(2(2
)83)(4)(2(
82
)83)(4)(2(
83
)83)(82(
)83)(83(
82
83)(
2
222
2
2
22
22
22
2
2
xx
x
x
xxx
x
xxxx
xx
xxxx
x
xxxx
xx
xxxx
xxxx
xx
xxxf
:
3
1
4
4
6
2
)2823(
)22(
)42(
2
)83(
)2(lim
)4(
2lim)(lim
22222
xx
x
xxf
xxx
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2) , , ox :
i) 4,
4,
1
27
6
)(
x
x
x
xx
x
xf , 4ox
ii) 2
8)(
2
x
xxxg , 4ox
iii) 4
168)(
2
x
xxxh , 4ox
:
i) :
3
2
74
64
7
6lim)(lim
44
x
xxf
xx
3
2
14
24
1
2lim)(lim
44
x
xxf
xx
3
2)(lim)(lim
44
xfxf
xx :
3
2)(lim
4
xf
x
ii) ),2()2,0[ gA :
)42(2
)42)(2(
2
22)2(
2
8
2
8
2
8)(
223
42
xxxx
xxxx
x
xxxx
x
xx
x
xx
x
xxxg
: 842)4424(4)42(limlim)(lim444
xxxxgxxx
iii) :
4,
4,
1
1
4
4
4
)4(
4
168)(
22
x
x
x
x
x
x
x
xxxh
1)(lim4
xhx
1)(lim4
xhx
)(xh ,
x 4.
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1.4)
,
.
gf , h :
i. )()()( xhxgxf ox .
ii. lxhxfoo xxxx
)(lim)(lim
: lxgoxx
)(lim
4
5)4()(
2
2
xxxf
),2()2,2()2,( A . )(lim2
xfx
.
:
Ax :
)4(4
5)4(
4
5)4()( 2
2
2
2
2
xx
xx
xxf
Ax :
)4()()4()4()( 222 xxfxxxf
0)4(lim 22
xx
0)4(lim 22
xx
0)(lim2
xfx
(
).
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1.5)
:
Rx xx .
( 0x , 0x
).
( ).
:
) oxx
xxo
lim
) oxx
xxo
lim
( ).
:
)
i. 1lim0
x
x
x
ii. 1lim0
ax
ax
x
, ( *Ra )
)
i. 01
lim0
x
x
x
ii. 01
lim0
x
x
x
, ( *Ra )
)ii) )ii) )i) )i) ,
uax ).
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1.6)
))((lim))((lim xgfxfogoo xxxx
.
uxg )( oxx
uxgo
)(lim ( ).
lufouu
)(lim ( ).
, ouxg )( ox , : )(lim))((lim ufxgfoo uuxx
))((lim xfogoxx
ouxg )( ox .
:
i) x
x
x 3
5lim
0
ii) 393
3lim
0 x
x
x
iii) x
x
x 2
1lim
:
i) : xx
x
x
x
xx
x
x
x
53
5
3
3
5
5
35
5
3
5
ux 5 vx 3 05limlim00
xuxu
03limlim00
xvxv
.
:
3
5
03
5
1
11
3
5lim
1limlim
3
5limlimlim
53
5lim
3
3lim
5
5lim
53
5
3
3
5
5lim
3
5lim
000000
00000
u
v
vu
u
uv
v
u
u
xx
x
x
x
xx
x
x
x
x
x
uvuuvu
xxxxx
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21 : - http://sciencephysics4all.weebly.com/
sciencephysics4all.weebly.com
ii) :
x
xx
x
xx
xx
xx
x
x
3
)393(3
993
)393(3
)393)(393(
)393(3
393
3
ux 3 03limlim00
xuxu
.
:
61)39(
lim)393(lim3
3lim)393(lim
393
3lim
00000
u
ux
x
xx
x
x
uxxxx
iii) :
)1(
1
)1)(1(
1
1
1122 xxx
x
x
x
x
x
:
2
1
)1(
1
)1(
1lim
1lim
2
xx
x
xx
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1) f R
53)(2 xxf Rx . )(lim5
xfx
.
:
Rx :
2
53)(
2
53
53)(253
53)(2553)(2
xxf
x
xxfx
xxfxxxf
: 2
3
2
53lim
5
x
x
2
3
2
53lim
5
x
x ( )
2
3)(lim
5
xf
x.
2) x
vxxx
x
...2lim
0.
:
:
vx
vxv
x
x
x
x
x
vxxx
...
2
22
...2.
:
2
)1(...211...121
lim...2
2lim2lim
...2lim
0000
vvvv
vx
vxv
x
x
x
x
x
vxxx
xxxx
( 11 a 1 ).
3) : 1,
1,
9)42(3
)14(9)(
232
x
x
xa
xxaxf
.
,a )(lim1
xfx
.
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23 : - http://sciencephysics4all.weebly.com/
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:
:
149))14(9(lim)(lim 2223211
axxaxfxx
91269)42(39)42(3lim)(lim11
aaxaxfxx
)(lim1
xfx
:
2
3
3
1032013
0)32()13(0)9124()169(
9126149)(lim)(lim
2222
22
11
aa
aaa
aaxfxfxx
4) 6
62)(
x
axxf . Ra ,
Rxfx
)(lim6
.
:
Rlxfx
)(lim6
: )()6(626
62)( xfxax
x
axxf
. :
243612
6120612)]()6[(lim62lim66
aa
alaxfxaxxx
5) f 6)(lim3
xfx
6)(
819)(3)()(
xf
xfxfxg . )(lim
3xg
x.
:
:
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519)(1
33)(
1
519)()6)((
2519)(
33)()6)((
93)(
519)()6)((
519)(519)(
33)()6)((
33)(33)(
6)(
519)(
6)(
33)(
6)(
819)(3)()(
xfxf
xfxf
xf
xfxf
xf
xfxf
xfxf
xfxf
xfxf
xf
xf
xf
xf
xf
xfxfxg
:
15
4
60
16
10
1
6
1
5196
1
336
1
519)(
1lim
33)(
1lim)(lim
333
xfxf
xgxxx
6) f R :
i) Rlxfx
)(lim2
ii) )2()()2( xxfx Rx
l .
:
Rx )2()()2( xxfx .
20)2( xx )2(
)2()(
x
xxf
)2(
)2(lim)(lim
22
x
xxf
xx
. 1lim
)2(
)2(lim
02
u
u
x
x
ux
( 2 xu ).
: 1)(lim2
xfx
(1)
20)2( xx )2(
)2()(
x
xxf
1lim)2(
)2(lim)(lim
022
u
u
x
xxf
uxx
. ( 2 xu ). : 1)(lim
2
xf
x (2)
(1) (2) : 1)(lim2
xfx
1l .
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7) 6
3525lim
3
6
x
xx
x.
:
ux 6 5 :
3336 55 uxux ,
23226 55 uxux
555 66666 uxuxux
:
6
7
1
33lim
)1)(1(
)33)(1(lim
)1)(1(
)1)(1(2)1)(1(lim
1
)1(2)1(lim
65
32lim
6
3525lim
2345
2
1
2345
2
12345
2
1
6
23
16
23
1
3
6
uuuuu
uu
uuuuuu
uuu
uuuuuu
uuuuu
u
uu
u
uu
x
xx
u
uu
uux
8) RRf : 0)(lim1
xfx
.
2)(2)(3
)(3)(4lim
1
xfxf
xfxf
x
.
:
uxf )( :
u
uu
u
u
uu
u
uu
uu
uu
xfxf
xfxf
123
34
123
34
223
34
2)(2)(3
)(3)(4
:
3
1
023
134
1lim23
lim34
123
34
lim2)(2)(3
)(3)(4lim
0
0
01
u
uu
u
u
uu
u
xfxf
xfxf
u
u
ux
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9) 32
5)2()(
2
2
xxxxxf , )(lim
1xf
x.
:
2
31032 2 xxxx .
),1()1,2
3()
2
3,( fA
fAx :
)2(1)2(
32
5)2(
32
5)2()(
22
2
2
2
2
xxxx
xxxx
xxxxxf
:
)2()()2()2()( 222 xxxfxxxxxf
uxx 22 :
00lim)2(lim0
2
1
uxx
ux
00lim)2(lim0
2
1
uxx
ux
0)(lim1
xfx
.
10) f R
)()(2 xxfxf Rx . )(lim0
xfx
.
:
Rx :
2)(
2
)(2)(2)(2
))(2()(22)(2
0)(4)(40)()()()(
22222
222
xxxf
xx
xxxfxxxxxfxxxxf
xxxfxxxxfxf
xxfxfxxfxfxxfxf
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02
lim0
xx
x 0
2lim
0
xx
x 0)(lim
0
xf
x
11) x 1 : )(0)()(3 xgxgxf
0)(3)(lim1
xfxgx
, : 0)(lim)(lim11
xgxfxx
.
:
x 1 : )(0)()(3 xgxgxf .
x 1 :
)(0 xg )(3)()()()()(3 xfxgxgxgxgxf
)(3)()(0 xfxgxg ( x 1) 0)(3)(lim1
xfxgx
0)(lim1
xgx
.
:
000)(lim)(3)(lim3
1)()(3)(lim
3
1)(lim
1111
xgxfxgxgxfxgxf
xxxx
: 0)(lim)(lim11
xgxfxx
12)
x
xx
1lim
0 .
:
*Rx :
xx
xx
x 11
( 11
x )
:
xx
xxxx
x 11
0lim0
xx
0lim0
xx
01
lim0
x
xx
.