mits rusia
DESCRIPTION
jsklhnjhbTRANSCRIPT
-
..
( 221700
, 111100 , 110900 ,
080200 , 111900 - , )
2013
-
2
51(075.5) 22.1+74.4p - 33
20 2013 ., 3 .
: . , .. , ..
. , . .-. . ..
..
-33 . ( 221700 , 111100 , 110900 , 080200 , 111900 - ) / .., ... : , 2012. 34 .
.
51(075.5) 22.1+74.4p
.., .., 2013 .. , 2013.
-
3
....................................................................................................................... 4
1. ........................... 6
2. ...................................................................................... 7
3. ............................................................................................. 9
3.1. ..................................................... 9
3.2. .................................................................... 10
3.2.1.
......................................................................... 10
3.2.2. 0
0 .......................................................................... 11
3.2.3. , 0 .......................................................... 13
3.2.5. - 1 ............................... 14
4. ..................................................... 15
5. .............................................................................. 16
6. ......................................................... 18
8. ................................................................................................. 21
9. ............................................................................................................. 22
10. ............................................................................................... 24
11. ........................................................................................ 30
12. ............................................................. 31
13. ................................................................................ 34
-
4
,
.
,
: , ,
,
,
.
,
, ,
.
,
,
.
-
5
c Ra,A)x(flimax
)f(Da )f(Da
-
00 0,
1
-
x -
1x
-
ex
x
x
11lim
e
1lim
0
1sin
lim0
x
x
x
1sin
lim0
-
00
)x(fy ax 0 ( )y(Da,Ra )y(Da )
I II
)0()(lim0
afxfax
,
)0()(lim0
afxfax
)()0()0( afafaf
)0( af , )0( af
)0( af
)0( af
( )
)()0()0( afafaf a
)0()0( afaf a
-
6
1.
n , N ,
)(nfxn nx .
a nx ,
1N , axn 1Nn . :
axnn
lim . (1)
1. , ,
22
2lim
3
3
n
n
n.
1. 2
2
23
3
n
n,
22
2)(:)(0
3
3
n
nNnN .
2. , n
22
23
3
n
n, ..
n .
3. 3 24)( Nn . , 2
2
23
3
n
n.
.
, ,
aann
lim .
1. .3
2,
13
22
a
n
nan 2. .2,
32
24
a
n
nan 3. .
2
3,
12
23
a
n
nan
4. .3
5,
13
25
a
n
nan 5. .5,
1
25
a
n
nan 6. .4,
2
142
2
a
n
nan
-
7
7. .1,1
33
3
a
n
nan 8. .3,
12
26
a
n
nan 9. .2,
41
832
2
a
n
nan
10. .3,1
3
a
n
nan 11. .3,
6
31
a
n
nan 12. .2,
5
32
a
n
nan
13. .4
3,
14
232
2
a
n
nan 14. .
5
3,
54
322
2
a
n
nan 15. .4,
2
423
a
n
nan
2.
A )(xf ax ,
0 0 , Axf )( ax0 .
:
Axfan
)(lim . (2)
Axfn
)(lim , Nx .
)(lim xfan
, Mxf )( ax0 ,
M .
)(xf ax .
0)(lim
xan , )(x ax .
ax ax , 0 ax ; ax ax
0 ax .
)(lim)0(0
xfafan
)(lim)0(0
xfafan
(3)
)x(f a .
)x(f ax
, )0()0( afaf .
2. , 831
1215lim
2
31
x
xx
x.
1. 8 31
1215)(
2
x
xxxf 31x ,
-
8
8
31
1215)(
3
10:0)(0
2
x
xxx .
2. )( , M ,
8
31
1215 2
x
xxMx ,
..
8
31
1215 2
x
xx.
)( ,
8
31
1215)(
3
10
2
x
xxMxx
3. :
153
1
153
183158
31
1215 2
xx
x
xx
(.. 31x , .. 031 x ,
31x ). ,
8
31
1215
153
1,
3
1
3
1,
153
1 2
x
xxMx .
, 15
,
)(3
10 x
8
31
1215
153
1,
3
1
3
1,
153
1 2
x
xxx ,
..
831
1215lim
2
31
x
xx
x.
.
, .
1. .13
15112lim
2
3
x
xx
x 2. .10
2
823lim
2
2
x
xx
x 3. .1
21
156lim
2
21
x
xx
x
4. .631
3129lim
2
31
x
xx
x 5. .5
1
23lim
2
1
x
xx
x 6. .5
3
6lim
2
3
x
xx
x
-
9
7. .31
396lim
2
1
x
xx
x 8. .9
3
9154lim
2
3
x
xx
x 9. .16
3
6206lim
2
3
x
xx
x
10. .71
235lim
2
1
x
xx
x 11. .3
21
252lim
2
21
x
xx
x 12. .19
6
6173lim
2
6
x
xx
x
13. .1931
6173lim
2
31
x
xx
x 14. .8
51
1215lim
2
51
x
xx
x 15. .8
31
1215lim
2
31
x
xx
x
3.
3.1.
,
.
. )(lim0
xfxx
)(lim xfx
.
, )(lim xf .
.
)(lim xu )(lim xv , )x(uy )x(vy
x ,
1) vuvu limlim)lim( ; 2) vuuv limlim)lim( ;
3) v
u
v
u
lim
limlim ( 0lim v );
4) ,lim,lim vu
)lim( vu ;
5) ,0lim,lim vu )lim(uv ; 6) vlim , 0lim
v
A
( constA );
7) ,0lim v v
Alim
( 0 constA ); 8) 0lim vu , vv lim;1lim0 ;
9) vulim , vv lim;1lim .
.
. , ,
,
, ,
.
-
10
3. .xarctglimx
5
3
5
.3
355
3
5
3lim
5
arctgarctgxarctg
x
, 50 x .
, :
3.2.
,
1,,0,,0,,
0
0 00 .
.
.
3.2.1.
,
, :
1) x
;
2) :
1x ( x 0 ).
4. xx
xlimx 43
122
2
.
1 : 3
2
43
12
43
12 2
2
2
x
xlimxx
xlim
xx, x
2
1
x
x
4 .
-
11
2 : 3
2
43
2
43
12
0
1
43
12 2
2
2
02
2
lim
x,x
xx
xlimx
.
.
1. xx
x
x 43
12lim
2
2
. 2.
3
3
21
75lim
x
xx
x
. 3.
13
6lim
x
xx
x.
4. 1
1lim
2
3
x
x
x. 5.
12
12lim
2
x
x
x. 6.
54
3lim
24
3
xx
xx
x.
7. 142
235lim
2
2
xx
xx
x. 8.
13
13lim
2
x
x
x. 9.
37
13lim
xx
x
x
.
10. 32
32lim
6
3
xx
x
x. 11.
5
53lim
3
23
x
xxx
x. 12.
13
74lim
2
2
x
xx
x.
13. 65
8lim
2
3
xx
x
x. 14.
xxx
xx
x
36
25 17lim . 15.
1
432lim
23
23
xx
xxx
x.
3.2.2. 0
0
) - . :
)( ax ,
;
5. 1
123
23
1
xxx
xxxlimx
.
:
0
0
1
123
23
1 xxx
xxxlimx
)1()1(
)1()1(lim
2
2
1 xxx
xxx
x 2
2
1 11
11
)x()x(
)x()x(limx
0
2
0
)1(
)1(lim
1
x
x
x.
) - .
:
1) ,
, ;
2) .
-
12
.
6. x
xlimx
11
0
.
0
011
0 x
xlimx
11
1111lim
0 x
x
x
x
x
11
11lim
22
0 xx
x
x
11
11lim
0 xx
x
x
11lim
0 xx
x
x 2
1
11
1
11
1lim
0
xx.
7. 1
13
1
x
xlimx
.
110
0
1
1 23363
1 tx
,tx,tx,tx
x
xlimx
1
1lim
3
2
1 t
t
t
)tt()t(
)t()t(limt 11
1121
3
2
1
121
tt
tlimt
.
) ,
, .
,
, .
:
1sin
lim0
x
x
x, 1
0
sinlim (3)
8. 20
21
x
xcoslimx
: xsinxcos 2212 ,
22
0
0212
2
020
x
xsinlim
x
xcoslim
xx.
.
9. xsin
tgxlimx 2
.
2
1
2
11
20
0
2 22
xcos
limxsin
limxcosxsin
xsinlim
xsin
tgxlim
xxxx.
-
13
.
1. 12
1lim
2
2
1
xx
x
x. 2.
x
xx
x
11lim
0. 3.
x
x
x
4sinlim
0.
4. xxx
xx
x 34
)32(lim
23
22
3
. 5.
12
1lim
2
3
1
xx
x
x. 6.
x
x
x
2sinlim
0.
7. 256
35lim
216
x
x
x. 8. .
sin
2cos1lim
0 xx
x
x
9.
131lim
0 x
x
x.
10. .2sin
lim2
2
0 x
x
x 11.
4
2lim
4
16
x
x
x. 12.
x
tgxtgx
x 2sin
11lim
.
13. x
mx
x
11lim
3
0
. 14. .
2cos
cossinlim
4 x
xx
x
. 15.
ax
xax
ax
lim .
3.2.3. , 0
0
0
.
10. xxxx
5lim 2
.
, ,
,
x :
xxxlimx
52
xxx
xxxxxxlimx 5
55
2
22
xxx
x
x 5
5lim
2
2
5
11
5
511
5lim
x
x.
0
0
0
: 0)(,)( xgxf ax .
)(1
)()()(
xf
xgxgxf
)(1
)()()(
xg
xfxgxf
0
0
.
11. 2
)1(lim1
xtgx
x
.
-
14
,
.
02
11
xtg)x(lim
x
2cos
2sin)1(
lim1 x
xx
x
2cos
1lim
2sinlim
11 x
xx
xx
22sin
1lim1
1 x
x
x
21
2
12
sin
)1(2lim
2
1
x
x
x.
, .
x1 ,
2
)1(lim1
xtgx
x
22lim
0
tg
2lim
0
ctg
2sin
2cos
lim0
2
coslim0
2sin
lim0
2
2sin
2lim2
10
.
.
1.
4
4
2
1lim
22 xxx. 2. tgx
x)
21(lim
2
. 3. xxxxx
22 1lim .
4. tgxxx
4sinlim2
. 5.
xtgx
x cos
1lim
2. 6. xtgx
x2
4lim
4
.
7. xxxx
3lim 2 . 8. xxx
lnlim0
. 9.
1
2
1
1lim
21 xxx.
10. ctgxe xx
)1(lim 20
. 11. ctgxxx
2sinlim0
. 12. arcctgxxx
lim .
13.
8
12
2
1lim
32 xxx. 14.
arctgxx
x 2lim
. 15. xxxx
x
22 2lim .
3.2.5. - 1
- 1 ( e )
.
-
15
n
n
11 n
. :
...7182,21
1lim
e
n
n
n (4)
n1 , elim
1
01 ,
.
12. x
x x
xlim
2
22
2211
2e
xlim
x
xlim
x
x
x
x
.
.
1. x
x x
51lim . 2.
x
x x
3
11lim . 3. x
xx
1
021lim
. 4. x
xx
1
021lim
.
5. x
x x
x
1lim . 6. x
x
xx
1
041lim . 7. x
x
xx
1
041lim . 8.
x
x x
x2
12
12lim
.
9. xx
x 21
031lim
. 10.
x
x x
xlim
2
13
23
. 11.
x
x x
x4
34
54lim
. 12. x
x
xx
2
021lim .
13. 3
41lim
x
x x. 14.
23lim
x
x x
x
. 15. x
xx 3
2
051lim
. 16.
x
x x
xlim
1
23
4.
:
mx
xa
x
a
x
x m
x
x
xx
1)1(lim,ln
1lim,1
)1ln(lim
000.
.
1. 1
1lim
3
4
1
x
x
x. 2.
923
3sinlim
0 x
x
x. 3. axxaxx
x
22lim .
4. 21 1
)1sin(lim
x
x
x
. 5.
x
x
x 2
)1ln(lim
0
. 6.
231 1
2
1
3lim
xxx.
-
16
7. 1
12
52lim
x
x x
x. 8.
)11(
cos1lim
0
xx
x
x. 9.
x
e x
x
1lim
0
.
10. x
a x
x
1lim
2
0
. 11.
121lim
40 x
x
x. 12.
x
x
x 5
11lim
3
0
.
13. x
xx
x 51
sinlim
. 14.
x
x
x 5sin
24lim
0
. 15.
932
8118lim
2
24
3
xx
xx
x.
5.
)(x )(x ax .
1. 0lim
ax
, ,
. )( o .
2. max
lim , m , , ,
. , 1lim
ax
,
. ~
, .
, , 0lim . ,
, .. )( o .
3. k , 0k ,
, k .
:
10.
, .. , )( o )( o .
20. ,
, .. )( o )( o , ~ .
30. ,
, .. ,lim max
~ ,1 ~ 1 , max
11lim .
-
17
:
0x ,
xsin ~ x , tgx~ x , arctgx ~ x , )1ln( x ~ x .
1. t .
3252 23,25 tttt .
3
5
23
25lim
23
25lim
3
032
52
0
t
t
tt
tt
tt.
, ,
.
2. 3sin
lim3sin3
lim)sin31ln(
lim02020
x
x
x
xx
tgx
xx
xxx.
:
)sin31ln( xx ~ xxsin3 , 2tgx ~ 2x .
.
1. : 1) xcos1 ; 2) xtgx sin
x . , x
xcos1 ,
xtgx sin .
2. 0x , : 1) x10 ; 2) 3x ; 3) 5
xtg ;
4) xxe ; 5) x3 , x , , x , , x
?
3. : 1) tt 22 sin tgtt , 0t ; 2) 1 xa
ax ln , 0x .
4. : 1) 54sin2 xx ; 2) 42sin xx ; 3)
11 3 x x .
:
5. 20
5sinlim
xx
x
x . 6.
xtg
xx
x
2lim
0. 7.
20 52sin
limxarctg
xx
x.
-
18
8. )21(ln
3sinlim
2
2
0 x
x
x . 9.
xe x
x 41ln
1lim
2
0
.
10.
xxtg
x
xxarctg
xtg
x 5arcsin
12sin
31
lim
3
3
0
11. 20 1lncosln
limx
x
x . 12.
3sin
2lim
2
2
0 x
xtg
x.
13. tgbx
xax
x
2
0
sinlim
. 14.
3
2
0 2sin
sin3lim
xx
xx
x
. 15.
2
2
0 5
4sinlim
x
x
x.
6.
)(xfy a ,
xax
yafxf )()( , ..
yx
0
lim = 0)()(lim0
afxafx
.
:
)(xf a , ax
, ..
)()(lim afxfax
.
)(xf a
:
1) ,
a (.. a );
2)
)(lim)(lim00
xfxfaxax
;
3) )(af .
a ,
, ,
.
-
19
)0()(lim0
afxfax
)0()(lim0
afxfax
,
)0(),0(),( afafaf , a
I .
I , ,
( )()0()0( afafaf , ..
a ,
) ( )0()0( afaf , ..
a );
)0()0( afaf a .
, I ,
II . II
.
1. 4
x
xy 4x
II , 4x
, , ..
4
lim04 x
x
x,
4lim
04 x
x
x (. 1). . 1
2.
.5,2,72
,5,21,24
,10,2
)(
xx
xx
xx
x
)(x 0x .
1x 5,2x ,
. )(x
, , ,
,
x .
1x 5,2x :
) 22lim)(lim0101
xxxx
; 2)24(lim)(lim0101
xxxx
.
-
20
)(x 1x
212)1( .
, 1x :
1x
)1()(lim)(lim0101
xxxx
.
1x )(x .
) 1)24(lim)(lim05,205,2
xxxx
; 2)72(lim)(lim05,205,2
xxxx
.
,
, .. 2-
. , 5,2x
I .
1)1(2)(lim)(lim05,205,2
xxxx
(. .2). . 2
.
, )(xfy :
1. x
y6
. 2. )5)(1(
1
xxy . 3.
.2,0
,0,2
x
xx
y
4. 1
12
xx
y . 5.
.0,2
,0,sin
x
xx
x
y 6. 24
4
xy
.
7. xe
y
11
1. 8.
2
2
9 x
xy
. 9.
152
32
xx
xy .
10.
.2,4
,20,4
,2,0,2
2
x
xx
xx
y 11. tgxy . 12. x
y121
1
.
13. 31
sin
x
xy
. 14.
12
12)2(1
)2(1
x
x
y . 15.
.2,3
,2,5,2
,2,5,0 2
x
x
xx
y
-
21
8.
10.
0
0. . 0lim
xf
ax
0lim
xax ,
xxf
x
xf
axax
limlim , .
20.
. .
xf
axlim
xaxlim ,
xxf
x
xf
axax
limlim , .
30. 1,,,0 00
0
0
.
,
0
0
, .
1.
152162
825lim
0
0
152162
825lim
234
23
1234
23
1xxxx
xxx
xxxx
xxx
xx
8
5
21321614
211013
23264
2103lim
23
2
23
2
1
xxx
xx
x.
2.
2lim
2lim
2limlimlim
22
x
x
x
x
x
x
x
x
x
x
e
x
e
x
e
x
e
x
e.
( ).
.
1. x
x
x
3sinlim
0. 2.
x
x
x ln
1lim
1
. 3.
xtg
tgx
x 3lim
2. 4.
x
x
x 3cos
sin21lim
6
.
5. x
e x
x 2sin
1lim
1
. 6. xx
xlnlim
0 7.
3lim
x
e x
x . 8. 20
cos1lim
x
x
x
.
9.
20
1
sin
1lim
xxxx. 10. ctgxe
x
x)1(lim 2
0
. 11.
30lim
x
arctgxx
x
. 12. 22 )2(
sin1lim
ax
ax
ax.
13. )21ln(
1lim
2
0 x
e x
x
. 14.
x
tgx
x 2cos
1lim
4
15.
31 1
lnlim
x
x
x .
-
22
9.
,
.
ax )(xfy ,
)(lim xfax
)(lim xfax
.
by )(xfy ,
bxfx
)(lim bxfx
)(lim .
bkxy )(xfy ,
kxxfbx
xfk
xx
)(lim,
)(lim kxxfb
x
xfk
xx
)(lim,
)(lim .
)(xfy
)()( xbkxxfy , 0)( x , x
x , bkxy .
1. 22
23 11
1
xx
x
xxy
1 xy ( 0x ).
2. )2(3 xxy .
)0,( ),2( .
)2(lim 302
xxx
, 2x .
, )2(lim 3
xxx
)2(lim 3
xxx
.
, . :
1) x
xfk
x
)(lim1
x
xx
x
)2(lim
3
2
limx
x
x,1
21
1lim
xx
xkxfbx
11 )(lim
x
x
x
x 2lim
3
2
)2(lim
x
xxx
x
-
23
)2(2
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-
34
13.
) :
1. , .. : / .. , ..
. .: , 2004. 520 .
2. , .. : / .. . 3- .,
. .: , 2004. 400 .
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) :
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/ .. , .. , .. . .: , 2008. 816 .
2. , .. / . .. . 3- .,
. .: , 2005. 368 .
3. , .. : . /
.. , .. . .: , 2008. 654 .
4. , .. (
): . / ... 2- ., . .:
, 1994. 206.: .
5. , .. : .
/ .. . 15- . .: , 2008. 336 .