xi_matematica (in limba rusa)
DESCRIPTION
XI_Matematica (in Limba Rusa)TRANSCRIPT
-
2- ,
-
2Toate drepturile asupra acestei ediii aparin Editurii Prut Internaional.Reproducerea integral sau parial a textului sau a ilustraiilor din aceast carte este permis doarcu acordul scris al editurii.
Autori: Ion Achiri, doctor, confereniar universitar, IE (Capitolul 4)Vasile Ciobanu, doctor, confereniar universitar, USM (Capitolul 1)Petru Efros, doctor, confereniar universitar, USM (Capitolele 810)Valentin Garit, doctor, confereniar universitar, USM (Capitolele 810)Vasile Neagu, doctor habilitat, profesor universitar, USM (Capitolele 3, 5)Nicolae Prodan, doctor, confereniar universitar, USM (Capitolele 6, 7)Dumitru Taragan, doctor, confereniar universitar, USM (Capitolul 2)Anatol Topal, doctor, confereniar universitar, USM (Capitolele 6, 7)
Comisia de evaluare: Dorin Afanas, doctor, confereniar universitar, USTAndrei Corlat, doctor, confereniar universitar, AMAliona Pogreban, profesoar, grad didactic superior, Liceul Teoretic Gaudeamus, Chiinu
Traducere din limba romn: Ion Achiri, Petru Efros, Antonina Erhan, Valentin Garit, Nicolae ProdanRedactor: Tatiana RusuCorector: Lidia PaaCopert: Sergiu StanciuPaginare computerizat: Valentina Stratu Editura Prut Internaional, 2014 I. Achiri, V. Ciobanu, P. Efros, V. Garit, V. Neagu, N. Prodan, D. Taragan, A. Topal, 2014Editura Prut Internaional, str. Alba Iulia nr. 23, bl. 1 A, Chiinu, MD 2051Tel.: 75 18 74; tel./fax: 74 93 18; e-mail: [email protected]; www.edituraprut.md
Imprimat la F.E.-P. Tipografia Central. Comanda nr. 3536CZU 51(075.3)M 34
ISBN 978-9975-54-152-7
Manualul a fost aprobat prin ordinul Ministrului Educaiei al Republicii Moldovanr. 267 din 11 aprilie 2014.Lucrarea este elaborat conform curriculumului disciplinar i finanat din Fondul Special pentruManuale.
coala/Liceul ...........................................................Manualul nr. ..................
Anulde folosire
Numele i prenumeleelevului
Anulcolar
Aspectul manualuluila primire la returnare
12345
Dirigintele clasei va controla dac numele elevului este scris corect. Elevii nu vor face nici un fel de nsemnri n manual. Aspectul manualului (la primire i la returnare) se va aprecia: nou, bun, satisfctor, nesatisfctor.
Acest manual este proprietatea Ministerului Educaiei al Republicii Moldova.
-
3
, .
. - , . , *, . , , , .
. (, , . .) , , . - , .
, , , , , , - . : , , . - - , - .
, , , . , , , . . , , , , , . ; ; -. (*) .
-
4 : A , ; .
. , - , .
, . , .
, .
! ! , - . , .
-
51 .
1.1.
, .
A B R,
Aa Bb .ba Rc , bca .
1 Rx m
, .1+
-
1
6
. RX (), ,Rc )( cxcx Xx . c () X. RX , , , m, M , Mxm Xx . .
, : RX (), Rm Xx , ).( mxmx ( , , [ , ] Rm [] Xx : ,Rm Xx .)
. ,RX (), () . c () X, 1c , (), c, () X. , ,Xx cx ,1cc < , ,1cx , 1c .
. Xa ( ), ,RX - ( ) X, ax ( )ax Xx . : Xa max= ).min( Xa =
=
NnnA1 , ,1max =A Amin .
. ( ) - RX , , , X, .sup X ( ) RX , - , , - X, inf X.. Xinf= ,supX= },|{ XxxY = =Yinf .sup =Y1. N , .
, N . RQZ ,, , .
2.
= NnnX1 , 1n .110 < n
3. }|{sin R= xxA , 1sin1 x .Rx. 2, ,0inf XX = ,1sup XX = 3, ,1inf AA =
.1sup AA = , () RX .
-
7
, , , . (. 2), : () , ().
1. , (), , , () . RX , , Y
X, , }.,|{ yxXxyY = R X , Y ,yx ., YyXx
, c , Xx Yy
.,
yccxycx (1)
(1) , c X, , c X, , c X. , .sup Xc =
, . . X c .c , .cc Xx , . > Mx 3 ( ) RX , . m X , :1) Xx ; mx2) 0> Xx , . +< mx ()
. X (), +=Xsup
).(inf =X R=X =Rinf .sup +=R Rx -, .+
-
1
8
1.
= *11 NnnA .:
, 1110 111 n .N n . , 0> ,11 An
.111
> n , 2 . , .1sup =A
, .0inf =A .,011 *N nn 0 - A ( n = 1), .0inf =A , ,mininf AAA = .sup AA
2. .42
2
+
= Nnn
nA
a) , A .) A.:a) , 1
40 2
2
n .1112 +
=
n , .1sup =A
+n
n ,042
2
Nn , A0 , , .0inf =A. , . 0 1.1.2. .
, .
. RE . - ( ) .: * Ef N Nn -
.)( Enf
-
9
f ,N . .
, )(nf nx :.)( 1nnx nx n- ,
.. 1. f N, , ,)( 0nnx
},1...,,1,0{\ kN .)( knnx 2. : 111 ,)(,)(,)( nnnnnn cba
11 )(,)( nnnn . .1. ,1,)( 1 nxx nnn =
, - .
2. ,,)( 0 naa nnn = .3. ,2,)( 2 = nbb nnn ...,2...,,2,1,0 n , ,
. :1) , , ., ,)( 1nnx
,)1(1 nnx += ,01 =x ,22 =x ,03 =x ,24 =x ...2) , 2, 3, 5, 7, 11, 13, 17, 19, 23, ...3) .
, - .
1. 21 =x ,21 nn xx +=+
,1n ,21 =x ,222 +=x ...,2223 ++=x2. 1,1 10 == xx 21 += nnn xxx .2n
: ,2,1,1 01210 =+=== xxxxx...,8,5,3 345234123 =+==+==+= xxxxxxxxx
, 1, 1, 2, 3, 5, 8, ..., 1.
, :
+
=
++ 11
251
251
51
nn
nx .Nn ()1 () (11751250) .
-
1
10
: -, , . (, , 3, - , , 4, 3, , 15, 10 . .).
. 1)( nnx 1)( nny , nn yx = .Nn
, 1
1
2)1(1
+n
n
1, 0, 1, 0, ... , - 1, 0, 1, 0, ... 0, 1, 0, 1, ... , , : }.1,0{
. 1)( nnx , nn xx =+1 .Nn
1)( nnx , 31 =x
,1,61 +=+ nxx nn : ...,3,3,3 321 === xxx1.3. .
* . 1)( nnx (- ), 1+ nn xx ( )1+ nn xx .*Nn 1)( nnx ( ), 1+< nn xx ( )1+> nn xx .*Nn . .. , ., ,1:)1(,)( 11 == xxx nnnn
...,1,1 32 == xx , 1)( nnx -
, :1. :
,,01 + Nnxx nn 1)( nnx ; ,,01 + Nnxx nn 1)( nnx .
2. , : ,,0 > Nnxn ,,11 + Nnx
xn
n 1)( nnx ;
,,0 > Nnxn ,,11 + Nnxx
n
n 1)( nnx . > (
-
11
,)( 1nnx :a) ;2
1++
= nnxn ) .)1(
1+
= nnxn
:a)
)2)(3()3)(1()2)(2(
21
32
21
2)1(1)1(
1+ =++++++
=
++
++
=
++
++++
= nnnnnn
nn
nn
nn
nnxx nn
.,0)2)(3(1
)2)(3(3444 22 >
++=
++++
= Nnnnnnnnnn
, ,,1 + > Nnxx nn , .) , .,0 *N> nxn .,12)1()2)(1(
1)1(
1:)2)(1(11 + M NMn ,
.|| MxMn
>
1. ,32
12,)( 1 ++
= nnxx nnn
:,,032
12 >++
= Nnnnxn , .
, . , .1,132
21322)32(
322212
3212 nb : .11 + = nnn bbb
2.2.
, IV . . . , , , 2 , 4 , 8 . . 64- . . ?
-
1
18
. 112 + = nnn bbb ( 11|| + = nnn bbb ) 1)( nnb . ., 7. , , , .. , 7.2.2.2. 1b 1)( nnb q .
:,12 qbb =
,)( 21123 qbqqbqbb ===,)( 31
2134 qbqqbqbb ===
...................................
8. 1)( nnb :.11
=n
n qbb (4)
. )(nP (4).1. ,1=n )1(P .2. )(kP ,1k .11 = kk qbb, ).1( +kP, .)( 1111 kkkk qbqqbqbb === +3. , )(nP
n. . 1)( nnb q ,1,1 =+ nqbb nn .1b2.2.3. n 1)( nnb , 1b q
.. , nbbb ...,,, 21 , , :
,11 nknk bbbb = + , , , . n :
nn bbbS +++= ...21 . (5)
-
19
, nS , :1) ;1=q .1 nbSn =2) .1q (5) q :
.121 qbqbqbqbqS nnn ++++= ,21 bqb = .,.. ,32 bqb = ,1 nn bqb = ,
....32 qbbbbqS nnn ++++= (6) (5) (6), :
= 1bqbSqS nnn .)1( 1bqbqS nn =
,1q .1111
qqbb
qbqbS nnn
=
=
: ,11
qqbbS nn
= 1q n - .)( 1nnb
,1)1(1
qqbS
n
n
= 1q (7) n 1)( nnb , , 1b q.. (7). . , ,
, . . .2...2221 6332 +++++: .2,2,1 63641 === bqb 370955161518446744071212
122 646364 ==
=S .
. , 30000000 - 1 , , .(: 20122013 700000000 , 614 .)
, : 1,01 >> qb 10,01
-
1
20
A
1. :a) 3, 3, 3, 3, ; ) ...; ,8
1 ,41 ,2
1 ,1 ) ...; ,271 ,9
1 ,31 ) 1, 9, 25, 49, 81,
2. ,)( 1nna :a) ;2 ,71 == ra ) ;5 ,31 == ra ) ;3,0 ,3,11 == ra ) .5
1 ,72
21 == aa3. 1a ,)( 1nna :
a) ;12 ,13110 == ra ) .3 ,0200 == ra4. ,)( 1nnb :
a) ;21 ,101 == qb ) .3 ,2
11 == qb
n, nq ( ), ,1||
-
21
11. ,)( 1nnb 31
32
21=
++
bbbb
.52321 =++ bbb12. 1)( nna ,nS :
a) ,41 =a ,31
=r ;14=n ) ,53
1 =a ,71
=r .25=n
13. , a, b, c , ,2 bca ,2 acb abc 2 .
14. ,)( 1nnb :a) ,91 =b ;21 nn bb =+ ) ,101 =b .5
11 nn bb =+
15. ,)( 1nnb :a) ,124 =b ;16
7237 =b ) ,167
41 =+ bb .87
123 =+ bbb
16. , ,403 =S .606 =S .9S17. Rzyx ,, , :
a) zyx ,, ;) zayx ,, + ;) bzayx ++ ,, .
18. Rx 26,12,12 ++ xxx ?
19. ,)( 1nnb :a) ;3,124 == qS ) .2,16 == qS
20. R : .2801371 =++++ x21. ABC, , -
. 2?22. 180 ,
,1q , 36 , . .
5. , 5 , 3 . - , .
6. 10 . 100 , 20 , . ?
7. 9% . 5 , 2700 ?
8. 100 0,7. 26. , 14,8?
9. , , 150 , 60 , . , 12 ?
10. . 10 ?
-
1
22
3 . -,
, 1 - . .
3.1. Ra
),,( + aa .0> a ),( aU ).,( aV
, }.||{}|{),( .|| , Nn ,
0nn > , .|| 00 axn2. ,
, 0> . , : Ra - 1)( nnx , 0> *N n , nn >
,||
1. .1,)( 1 nxx nnn = , .0lim = nn x
1 (490 430 . .) - .
-
23
U 0, ).,( =U *Nn ,
,1
>n , .10 n ,
, ,11 +
=
n -
U 0., 0 .1,)( 1 nxx nnn = : .01lim =
nn
2. , .2112lim =
++
nn
n
, 0> *Nn , ,*Nn
,nn > .2112
, .1121
12 -
nn
.,111 *N+
=
nn
, 0> *Nn , , .2112lim =
++
nn
n
3. , ,)1(,)( 1 nnnn xx = . , ,Ra .)1(lim an
n=
,0> ,21
= Nn, nn > .2
1||
-
1
24
, ,0> , nn11
=
n ,
=
1
1n -
. , 0> *11 N
=
n , -
, ,01lim = nn
.0>
: .0,01lim >=
nn
. , , - ,)( 1nnx ,)1( nnx = .
. , 1)( nnx , +=
nn
xlim , 0> Nn , >nx .nn >
, 1)( nnx -, =
nn
xlim , 0> Nn , , 1)( nnx , =
nn
xlim , 0> Nn , - >|| nx .nn > . , +=
nn
xlim ,lim =
nnx .lim =
nn
x
1. ,)( 1nnx .2nxn = , .lim +=
nn
x2. ,)( 1nnx .2nnx = .lim =
nn
x3. ,)( 1nnx nx nn = )1( , .lim =
nn
x
,)( 1nnx .1,
,Nn nn > .|| >nq .0> >|| nq .|| >nq -
,1|||,| >qq :.loglog||log |||||| qq
nq nq >>
, 0> 1][log || += qn , >|| nq .nn > .lim =
n
nq
-
25
:
};0,|{),( >>=+ xxU R};0,|{),( >>= xxU R ,+ .
+ 0> . , . , , ,+ . , , : . , :
. , ( , ,+ ) - ,)( 1nnx ),( aU un , Uxn .unn > , , ., ( , , ), . 9. , .
10 (1). - . , 1)( nnx -
. 1+ nn xx , .*Nn -, }1{ nxn . .),(sup 00 R
N
=
xxx nn
, 0> n , .0 > xxn 1)( nnx, ,
>> 0xxx nn .nn > ,
, 0x , ,00 + +
-
1
26
: , 1)( nnx .
.22 1 nn xx + : 0)2)(1(2 222 1 >+=+=+ nnnnnn xxxxxx .Nn , .,22 1 + > Nnxx nn ,,0 > Nnxn , nn xx >+1 .Nn , .
, , - .
: ,221
-
27
3.4. e
14. ,11,)( 1n
nnn nxx
+= . 10 () 3.1 :
,...... 2121 n nn aaanaaa
+++
....,,, 21 +Rnaaa
, ,11n
n nx
+= ,1n . .)( 1nnx .1,11...,,11,11
444 3444 21
n
nnn +++
3.3. 13. ,...,)( 11111 +++= nnnn qbqbbSS - 1||0
-
1
28
, 1+n :
1112111
21111
111 111
nnn
nn
n
nnn
nnn
nnnn
+>
++
+>++
+>+
+
+ +++
,11111
1 nn
nn
+>
+
++
.1n
, ,1+< nn xx ,1n , .
, . 2+n .2
1,21,11...,,11,11
444 3444 21
n
nnn +++
2+n ,
+>+
++
++2
21
21112
21
2111
n
n
nnnn
,411411114
11122 2
+>
++
+nn
n
n
nnnnn .1n
, ,11n
n nx
+= ,1n . - 1)( nnx , , , .
,11,)( 1n
nnn nxx
+= 1, e. e
(2, 3). e - 1815 . 2. 1728 . 3 ,
...5907182818284,211lim ==
+
en
n
n
: .11lim enn
n=
+
(8)
. e - . - , , , ).log(ln xx e=
1 (17071783) , .2 (17681830) .3 (17001782) .
-
29
1. .73
83lim16 +
++
n
n nn
: 11 (8), :
=
+
+=
++=
++
++
+
+
+
)16(73
11
731616
7311lim73
11lim7383lim
nn
n
n
n
n
n
n nnnn
.7311lim 2
73)16(1
173
ennn
n
n=
+
+=
++
+
2. ,)( 1nnx :a) ;322 nnnxn += ) ;13
2+
+= n
nxn
) ;1nnnxn
+= ) .
31...3
1121...2
11
n
n
nx+++
+++=
:a) : =
++
=
++
+=+
nnnn
nnnnnnnnn
nnn 3232lim
32)32(lim)32(lim
22
222
.11321
32lim
2
=
++
=
nnn
nn
n
) .31
0301
13lim
21lim
13
21lim
13
21lim13
2lim =++
=
+
+=
+
+=
+
+=
++
n
n
n
n
nn
nn
nn
n
n
nnn
) , :.0
)1(1lim
)1()1(lim1lim =
++=
++
+=
+ nnnnnn
nnn
nnnnn
) n :
.34
311
211
lim34
311
311
211
211
lim
31...3
1121...2
11lim 1
1
1
1
=
=
=
+++
+++
+
+
+
+
n
n
nn
n
nn
n
n
-
1
30
A
1. 1)( nnx , :a) ;2
23n
nxn +
= ) ;6sin
= nxn
) .17)1( nxn
n +=
2. n- :a) ...;,6
5,43,3
2,21 ) 2, 4, 6, 8, 10, ...; ) 3, 3, 3, 3, ...; ) ...,81
1,271,9
1,31
3. : a) ; ) -; ) .
4. , 1)( nnx , .21 n
nx
=
5. 1)( nna , :a) 21 =a ;4=r ) 11 =a ;2=r ) 101 =a ;5=r ) 31 =a .7=r
6. 100 ,)( 1nna :a) ;5,21 == ra ) .1,11 == ra
7. ,Rx :a) ;)(,)(,1 2222 xaxax +++ ) .,, 22 xbxabxa +++
1. , .2. , , ,)( 1nnx
,)1( nnx = .3. , , :
a) ;414lim = n
nn
) ;212lim 22
=+
nn
n ) ;2
15432lim =
+
nn
n ) .51
65lim =++
nn
n
4. , , :a) ;2
111lim
+
nn
n) .115
12lim ++
nn
n
5. , ,)( 1nnx :a) ;1
12++
= nnxn ) ;3
11 nnx += ) .111+
+= nn nx6. :
a) ;11lim+ nn ) ;
2lim 2 nnn +) ;
35lim nn ) ;13
2lim+
+ n
nn
) ;2
lim2
nn
n
C ) ;22lim
n
n
) ;
54253lim 1+
++nn
nn
n) );32(lim 2 nnn
n++
) ;41...4
1121...2
11lim
n
n
n+++
+++
) ;1
1lim2n
n nn
+
) ;2...4321lim n
nn
++
) ;32
12lim 2
2
++
++ n
nnn
n) ;32
12limn
n nn
+
) .!)!1(
!lim nnn
n+
-
31
8. ,)( 1nnb :a) ;6,21 == qb ) ;2
1,101 == qb ) .2,31 == qb9. , 1)( nnx
, :a) ;3,2 11 nn xxx == + ) ;2,4 11 nn xxx +== + ) ;3
1,4 11 nn xxx == + ) .5,1 11 nn xxx +== + .
10. naaa ...,,, 21 .a) n ,nS ;2,23,5 1 === raan) 1a n, .88,2,18 === nn Sra
11. nbbb ...,,, 21 . ,nS :a) ;9,5,1280 1 === nbbn ) .8,2,384 === nqbn
12. 8 . 2 , . 60 ?
13. , , 4,9 , - 9,8 /. , 8 .
14. Rx , 13,1,22 22 + xxx - .
15. 1)( nnx , :a) ;)1(1 nnx += ) ;2
)2(2n
nn
nx+
= ) ;2sinnnxn = ) ;cos nxn = ) .3
)1(2n
nxn
n+
=
16. 2. .5
4 .17. , , :
a) ;2112lim =
++
nn
n) .0
112lim 2 =+
nn
n
18. , , :a) ;4
93
16lim + n
nn
) .1122lim
++
nn
n
19. , 729 , a , - . a . 6 , 64 . a.
20. , 4 1.
21. , ,,,)(,,)( 2222 R++ xaxaxaxa . n , 2)( xa + .
22. : a) ;1213lim
++
n
n
n) );12(lim 23 +
nn
n) );31(lim 5nn
n+
) ;3212lim n
nn
) ;13
221lim
++
nn
nn) ;...321lim 2n
nn
++++
) ;1limn
n nn
+) ;11lim
1+
+ nn n ) ).11(lim
22+
nn
n
-
1
32
1. 1)( nnx :.32)1( 1
+= nx nn
2. ,)( 1nnb ,41 =b .)3(1 nn bb =+3. 1)( nnx , -
:.
11 2 +
+=n
nxn
4. , ,)( 1nnx .)12(...31
)9(...1110
+= n
nxn
5. ,)( 1nna ,3
551 =+ aa .72
6543 = aa
6. ,)( 1nnb :,32
4524 = bb .512
40546 = bb
7. 26 .., 2 .. , . - 40 ..
9422 .. , .
1. 5 :)( 1nnxa) ;2
23+
= nnxn ) .3
)1(1 nnx
+=
2. 1)( nnx , :.12
12+
= nnxn
3. ,)( 1nna :,1642 =+ aa .2851 =aa
4. ,)( 1nnb :,412 = bb .813 = bb
5. , 3 .., , . , , 48 ..
: 45
A
: 90
-
33
,,
......
*2
12
1N
++
+n
aa
an
aa
an
nn
+
Rna
aa
...,
,,
21
- 1
n
nn yx
:
*1,
N
+n
xx
nn
:
*1,
N
+n
xx
nn
R
X
,
-.
M
-
X
,
:1)
X
x
;
Mx
2)
0>
X
x
,
.
>
M
x
R
X
,
.
m
X
,
:1)
X
x
;
mx
2)
0>
X
x
,
.
++= U 1=l ,Oy ,0),22,22( >+= V
20 =x Ox , Vx , Uxf )( (. 2.1).
2. ,}1{\: RR f ,11)(
2
= xxxf
1. - 1)( nnx , ,1nx 1nx ,n - 211
1)(2
+=
= nn
nn xx
xxf n (. 2.2).
3. ,: RRf
+
-
37
, 1 ( 2) , 1=l ( 2=l ) f 20 =x( 10 =x ), 3 , f .10 =x
. REf : )( RE R0x -
E. ( ). , f Rl x0 , U l V 0x , }){\( 0xEVx I ,
.)( Uxf
f 0x lxfxx = )(lim0 lxf )( 0xx : )(xf x, ,0x l, )(xf - l x, .0x
, l ),(}||{ += 0x
),(}||{ 000 += , , , V U. , , ).( = , .
( 1, ). , f Rl x0, 0> 0)( >= ,
}{\ 0xEx
-
2
38
. 2.4
y
xO x
l
f(x)
+lU
V
l
0x 0x +0x
)(xfy =
. ( )1. , f 0x + , 0> 0)( >= , }{\ 0xEx
xf : .)(lim0
+=
xfxx
,, 1 :+=
)(lim
0xf
xx )( 0 Rx , ,0> 0)( >= , },{\ 0xEx
.)(|| 0 > 0)( >= , }{\ 0xEx
xf : .)(lim0
=
xfxx
2 : =
)(lim0
xfxx
)( 0 Rx ,0> 0)( >= , },{\ 0xEx .|)(||| 0 > 0)( >= , ,Ex
.|)(| lxfx4. =
)(lim xf
x, ,0> 0)( >= , ,Ex
.)(|| xfx5. =
+)(lim xf
x, ,0> 0)( >= , ,Ex
.|)(| >> xfx
. 1. ,0x Rl f 0x : x, ,0x )(xf f l (. 2.4).2. ,: RN f ,)( nanf = - 3 5 ( ) - .3. , - 1)( nnx 1)( nnx },{\ 0xE
0limlim xxx nnnn == , 1))(( nnxf 1))(( nnxf , f
.0x 3 , f .0x4. , , .
-
39
, .
1. , ,
,: RR f
>+=
+= U .3=l ,1
-
2
40
1.3. REf : )( RE R0x -
E. , 0x ),( 0xEE = I ).,( 0 +=+ xEE I , x0 E.
R0x () E. x 0x ( ) 0xx < ( 0xx > ), 00 xx ( ).00 + xx 00 =x
0x ( ).0+x
,: RR f
+
-
41
. 2.5
y
xO
l
)0()( 00 = xfxl
0x
)(xf
l
)0()( 00 += xfxl
. , R= )( 0 xll f x0 ,R U l V 0x , EVx I , .)( Uxf . , R= )( 0 xll f x0 ,R U l V 0x , + EVx I , .)( Uxf )( 0 xl )( 0 xl -
f .0x :
),(lim)(0
00 xfxl
xxxx
=
:),(lim)0(
00 0xfxf
xx = )(lim)0(
00 0xfxf
xx +=+
(. 2.5). ,00 =x :
).0()(lim),0()(lim00
+==+
fxffxfxx
, - . , .
( ). , Rl f x0 ,R 0> 0)( >= , Ex += , Ex +
-
2
42
. ,)(lim
0lxf
xx=
-
lxf = )0( 0 .)0( 0 lxf =+. , ),0( 0 xf
)0( 0 +xf , .,)0()0( 00 R=+= llxfxf ( ) : lxf
xx=
)(lim(
00 ))(lim
00=
+lxf
xx
,0( > ,01 > 02 > , },{\ 0xEx 010 xxx
-
43
1. , 20 =x :RE
a) ;12 *
+
= NnnnE ) ;
2)1(2 *
+= NnE nn
) .1234 *
++
= NnnnE
2. :a) ;1
32)1(
++
= NnnnE n ) ;|))1(3(3
+
+= Nnn
nE n ) .3cos11
+
= NnnnnE
3. E, 0x , 0x E:
a) ),4,0[\R=E ;40 =x ) ,2)1( *
+
= NnnnE
n
}.1,1{0 x
4. , ,:a) ;2)1(lim
1=+
x
x) ;232
1lim2
=
x
x) ;2
141
23lim
21
=
+
xx
) ;3)21(lim1
=
xx
) ;1)3(lim2
=
xx
) .1lim 21
=
xx
5. :a) ;
1532lim 2
3
1 ++
xxx
x) ;
253252lim 2
2
2+++
xxxx
x) .
541lim 2
3
1
+ xx
xx
6. , , :a) ;1)352(lim 2
2=+
xx
x) ;13
73lim2
=
++
xx
x) .21
12lim =+
+ xx
x
7. , :a) ;1
1coslim1
xx ) ;sinlim
2 xx
) .sin1lim0 xxx
8. 0x :: 1 RDfa)
>++
=
,2,,2,12)( 2 xxx
xxxf ;20 =x ) ,1143)( 2
2
++=
xxxxf }.1,1{0 x
}.0,3{311 2 =+ aaa ,0=a ,1)(lim1)1()1(1
===
xfllx
,3=a .10)(lim10)1()1(
1===
xfll
x
, f 10 =x , : 1)1()1( 2 =+= all}.3,0{31 += aa 0=a ,1)(lim1)1()1(
1===
xfll
x 3=a
.10)(lim10)1()1(1
===
xfllx
, 0=a f , 1 1, }3,3{a f .
1 D .
-
2
44
2 .
2.1. , .
, .
E R, 0x E REgf :, , ,)(lim
0axf
xx=
,)(lim
0bxg
xx=
a b . : f 0x ,Rc fc
0x ).(lim])([lim00
xfcacxfcxxxx
== , - .
. , ,1)( =xf .lim0
ccxx
=
-, 0x .
9. , f 0x , ),(lim
0xfl
xx= :: RDf
a)
>+
=
,1,2,1,13)(
3 xxx
xxxf };1,0{0 x ) ,24)(
2
xxxf
= }.2,0{0 x
10. ,, Zkxk :: RR fa) ),sgn(sin)( xxf = ;, Z= kkxk ) ],[)( xxf = ., Z= kkxk
11. :: RR fa) ];[)( xxxf =) ];[cos)( xxf =) ),(sin)( xxf =
+
+=
,1,3,1,)1()(
2xax
xaxxf ;10 =x )
=
-
45
f , g 0x , gf 0x )(lim)(lim)]()([lim
000xgxfbaxgxf
xxxxxx == ()
() . 1)( nnx },{\ 0xE -
.lim 0xxnn = f g ,0x axf nn = )(lim .)(lim bxg nn =
, .)]()([lim baxgxf nnn =
, , .)]()([lim0
baxgxfxx
=
f , g ,0x gf 0x )(lim)(lim)]()([lim
000xgxfbaxgxf
xxxxxx ==
. ,...,,, 21 nfff -
.0x , , ,...21 ffff n ====, ,)](lim[)]([lim
00
n
xx
n
xxxfxf
= ,Nn .1n
f g 0x ,0)(lim0
=
bxgxx
)()(
xgxf 0x },{\ 0xE g
f
0x )(lim)(lim
)()(lim
0
0
0 xg
xf
ba
xgxf
xx
xx
xx
==
. f g 0x 0)( >xf Ex ,
,: REf g )()]([))(( xgg xfxf = 0x ( )00
.)](lim[)]([lim)(lim
)( 0
00
xg
xx
bxg
xx
xxxfaxf
==
- .
E F R, 0x E, ,: FEu RFf : , ,: REuf o )),(())(( xufxuf =o
Ex , . 1) ,)(lim 0
0uxu
xx=
2) 0)( uxu x 0x E ,0xx 3) ,)(lim
0luf
uu=
uf o 0x .)(lim))((lim00
lufxufuuxx
==
-
2
46
. 1. ),(lim))((lim00
ufxufuuxx
= , , ,
. , , )(xuu = , - , 1) 2) :
0uu .0uu 2. . , f g
0x gf
.0)(lim0
=
xgxx
, , ,,)(lim0
R=
aaxfxx
+=
)(lim0
xgxx
, gf + 0x .)]()([lim
0+=+
xgxf
xx
, :},{\,0)(,0)(lim 01
0xExaxf
xx>>=
;)(|| 10 +=> MM
, }{\ 0xEx
-
47
, ,lim 0
0xx
xx=
.0 Rx
, :1. ;2224232limlim4)lim(3)243(lim 2
22
2
2
2
2===
xxxxxxxx
2. ;1025)243(lim)3(lim)]243()3[(lim 222
2
2==+=+
xxxxxx
xxx
3. ;52
)3(lim)243(lim
3243lim
2
2
22
2=
+
=
+
xxx
xxx
x
x
x
4. ;322)243(lim)243(lim 5)3(lim2
2
)3(2
22][ ===
+
+
x
x
x
xxxxxx
5. 2)243(lim]2)3(4)3(3[lim 22
2
1==++
uuxx
ux 23( += xu cnd ).1x
1.
2. 00 3. 0 4. 5. 1 6. 00 7. 0
1. =+ a2. +=++ a)(
3. =+ a)(4. +=+++ )()(5. =+ )()(6. )0( = aa7. )0()( >+=+ aa8. )0()( >= aa9. )0()( = aa
20. )10(0
-
2
48
2.2. . 2.2. ,
. . .
I. ,: RR f ,)( nxxf = ,Nn 1n (. 2.6)
a) ,lim 00
nn
xxxx =
;0 Rx
)
+=
;,,,lim
n
nxnx
)
+=
;,,,lim
nnxn
x
) .lim +=+
n
xx
II. ,}0{\: RR f ,1)( n
n
xxxf == ,Nn 1n (. 2.7)
a) ,11lim00nnxx xx
=
};0{\0 Rx
) ;01lim = nx x
)
+=
;,,,1lim
0 n
nxnx
) ;1lim0
+=+ nx x
)
+=
.
,,,1lim
0 nn
xnx
III. -,: RR P ,...)( 110 n
nn axaxaxP +++= ,,0, niai =R ,00 a .Nn
a) ),()(lim 00
xPxPxx
=
;0 Rx ) ;lim)(lim 0 nxx xaxP =) ;lim)(lim 0 nxx xaxP ++ = ) .lim)(lim 0
n
xxxaxP
=
1. .32)1(4)1(3)243(lim 33
1=+=+
xx
x
2. .)3(lim)253(lim 22 ==+
xxxxx
3. .)2(lim)31002(lim 545 +==+
xxxxx
y
xO
,nxy =n
. 2.6
y
xO
,nxy =n
1=n 3
n
. 2.7
y
xO
n ,1nx
y =
y
xO
,1nxy =
n
-
49
. 2.9
y
xO
IV. P Q - :
nnn axaxaxP +++= ...)( 110 .,,0,0,...)( 00110 +++= NnmbabxbxbxQ mmm
,: REQP },0)({ = xQxE R .
a) ,)()(
)()(lim
0
0
0 xQxP
xQxP
xx=
;0 Ex
)
==
. ,0
, ,
, ,
lim)()(lim
0
0
0
0
mn
mnba
mn
xbxa
xQxP
m
n
xx
+x ,x ) mnx
xba
+lim
0
0 .
0)( 0 =xQ 4.1. .2
1226232422
26342lim 2
23
2
23
2=
++
=
++
xxxx
x
2. .21lim
42lim
24132lim 32
5
2
5
+=
=
=
+++
x
xx
xxxx
xxx
V. ),,0[),0[: ++f ,)( n xxf = ,Nn ,2n n (. 2.8)
a) ,lim 00
nn
xxxx =
);,0[0 +x
) ,lim +=+
n
xx n .
,: RR f ,)( n xxf = ,Nn ,3n n (. 2.9)
a) ,lim 00
nn
xxxx =
;0 Rx
) ,lim =
n
xx n ;
) ,lim +=+
n
xx n ;
) ,lim =
n
xx n .
. 2.8
y
xO
-
2
50
VI. ),,0(: +Rf ,)( xaxf = ,0>a 1a (. 2.10)
a) ,lim 00
xx
xxaa =
;0 Rx
) ,1>a : ,lim +=+
x
xa ;0lim =
x
xa
) ,10 a 1a (. 2.11)
a) ,logloglim 00
xx aaxx = ;00 >x
) ,1>a : ,loglim0
=+
xax ;loglim +=+ xax) ,10 x
) > 0 ,0lim0
=+
xx
;lim +=+
xx
) < 0 ,0lim =+
xx
.lim0
+=+
xx
IX. ],1,1[: Rf xxf sin)( = (. 2.13)
a) ;,sinsinlim 000
R=
xxxxx
) .sinlim,sinlim,sinlim xxxxxx +
],1,1[: Rf xxf cos)( = (. 2.14)
a) ,coscoslim 00
xxxx
=
;0 Rx
) .coslim,coslim,coslim xxxxxx +
. 2.11
y
xO
1>a
10 a10
-
51
,|2\: RZR
+ kkf xxf tg)( = (. 2.15)
a) ,tgtglim 00
xxxx
=
,20
kx k += ;Zk
) ,tglim0
+=
xkx
,tglim0
=+
xkx
.Zk
,}|{\: RZR kkf xxf ctg)( = (. 2.16)
a) ,ctgctglim 00
xxxx
=
,0 kx k = ;Zk) ,ctglim
0=
x
kx ,ctglim
0+=
+x
kx .Zk
X.
,2,2]1,1[:
f xxf arcsin)( = (. 2.17)
,arcsinarcsinlim 00
xxxx
=
].1,1[0 x
],,0[]1,1[: f xxf arccos)( = (. 2.18)
,arccosarccoslim 00
xxxx
=
].1,1[0 x
,2,2:
Rf xxf arctg)( = (. 2.19)
a) ,arctgarctglim 00
xxxx
=
;0 Rx
) ,2arctglim
=+
xx
;2arctglim
=
xx
) .arctglim xx
. 2.15
y
xO
2
2
23
. 2.16
y
xO2
2
23
2
. 2.17
y
xO
2
2
11
. 2.18
y
xO
2
11
. 2.19
y
xO
2
2
-
2
52
),,0(: Rf xxf arcctg)( = (. 2.20)
a) ,arcctgarcctglim 00
xxxx
=
;0 Rx
) ,0arcctglim =+
xx
;arcctglim =
xx
) .arcctglim xx
XI. ( )),,0[: +Rf ||)( xxf = (. 2.21)
a) |,|||lim 00
xxxx
=
;0 Rx
) ,||lim +=+
xx
,||lim +=
xx
.||lim +=
xx
, IXI. , , . , RDf : ,( RD D ) ),()(lim 00 xfxfxx = ,0 Dx .1. a) , xxxf x 2log32)( += , , ,
.124log324)4()log32(lim 24
24=+==+
fxx x
x
) ,.2ln22ln32
1ln6cos45ln36sinlncos4
5ln3)ln(sinlim 226
=+=
++
=
++
xxx
2. ;1lim 30 = xx ;02
1lim =
+
x
x ;lnlim
0=
+x
x ;lim3 =
x
x ;0lim 3
1
=
+x
x +=
+x
xctglim
0 etc.
. 2.20
y
xO
2
. 2.21
y
xO
1. :a) ;58
1lim 34
++
xxx
) ;1lim 3 24
++
xxxx) ;2lim 324
+
+x
xx
x
) ;12lim 33
2
++
xx
xx
) ;)31(lim 225 30 xxxx + ) ).31(lim 33
0xxxx +++
-
53
Exerciii propuse
2. :a) );103(lim 2
2+
xx
x) );352(lim 23 ++
xx
x) );132(lim 34 ++
xx
x
) );1005(lim 23 xxx
++
) ;13
1102lim 223
2 +++
xxxx
x) ;
532lim 32
3
xxxx
x
+
) ;32
lim 43
+
+ xxx
x) ;
1423lim 2
23
xxxx
x+
+
) .5
32lim 3423
0 xxxxxx
x
+
3. :a) );)3(log2(lim 25,02
xx
xx +
) );log(lim 30 x
x
x+
+ ) );2(loglim 5,00
x
xx
+
) );8422(lim2log
xxx
ex+
) );log(loglim 42
2exx
ex+
) ).lg(lim xex
x+
+
4. :a) );1(lim 4 4 +
x
x) );1)(1(lim 234 xxx
x+
+) );1)((lim 336 +
xxxx
x
) ;|1|1lim
1
+ x
xx
) ;|1|12lim
++
+ xx
x) .13
|1|lim
xx
x
5. :a) );tg3cos3(sinlim
6
xxxx
+
) );cosctg2(sinlim
4
xxxx
+
) ;),ctg3cos2(sinlim2
Z++
nxxxnx
) .),tgcos3sin2(lim Z+
nxxxnx
6. , :a) ,0sinlim)2sin(lim
...==
y
y
x
x ;2xy =
) ...,tglim)ln(tglim...1
==+
yxy
x
x ...=y
7. , :a) );21(coslim x
x/
+) ;sinlim 2x
x ) ),cos(sinlim xx
x/
+
/ .8. 0x :: RDf
a) ,||ln1)( xxf = };1,0,1{0 x ) ,)(
11
2
=xexf };1,1{0 x ) ,
21
1)(1
1++
=
x
xf .10 =x
9. Rm ,: RR f
>
+
=+
-
2
54
3
3.1. ,RE REgf :, R0x -
E. - .
1 ,)(lim0
axfxx
=
,Ra )( 0xV 0x , f .)( 0 ExV I
2 ,)(lim0
axfxx
=
,)(lim0
bxgxx
=
,, Rba ba < ( ),ba > )( 0xV 0x , )()( xgxf < (
))()( xgxf > }.{\)( 00 xExVx I. 2, =)(xg ),,( R Ex -
)( 0xV 0x , xf }.{\)( 00 xExVx I
0= , 0)( xf }.{\)( 00 xExVx I
3 . a) )(lim
0xf
xx ),(lim
0xg
xx
) )()( xgxf Ex 0x E, ).(lim)(lim
00xgxf
xxxx
. 3, ,)(lim0
=
xgxx
,)(lim0
=
xfxx
,)(lim
0+=
xf
xx .)(lim
0+=
xg
xx
4 . REhgf :,, :a) ,)(lim)(lim
00axgxf
xxxx==
,Ra
) )()()( xgxhxf Ex 0x E. .)(lim
0axh
xx=
11. , :
a) );sin21cos(lim6
xx
) );ln(sinlim 20
xx
) ;lim 4211
0xx
xe
) ;ln3lim 3 32 xxx
x+
) ;2sin2tglim
xx
) );cos(ctglim 30
xx
) ;)ln(cos2lim
sin
0 xx
x) .arcsin2lnlim0
x
x
-
55
:
a) );sin(lim xxx
+ ) ).(sinlim 2 x
xex
:a) Rx : .1sin1 x 1sin xxx (1). ,)1(lim +=
+x
x -
3 (1) , .)sin(lim +=+
xxx
) ,,1sin 2 R xeex xx ,)1(lim =
x
xe
3 , .)(sinlim 2 =
x
xex
. 2.22
y
xO D A
BC
x
3.2.
:
1sinlim0
= x
xx
a) ,11lim exx
x=
+
) ex x
x=+
1
0)1(lim
. |tg||||sin| xxx .22
-
2
56
.1. },0{\2,2
x 0sin x , ,0x -
, , |,sin| x :
-
57
) =
+
+
=
=+ +
+
xx
xe
xe
xx x
x
x
xx
)1ln()1ln(1lim1lim1)1(lim
)1ln(
0
)1ln(
00
,1ln)1ln(lim1lim00
==+
=
exx
ue
x
u
u )1ln( xu += 0u .0x
) .10cos11cos
1limsinlimcos1sinlimtglim
0000===
= xx
xxx
xxx
xxxx
) ,21sinlim2
1
2
2sinlim212sin2limcos1lim
2
0
2
02
2
020=
=
==
uu
x
x
x
x
xx
uxxx 02 =
xu
.0x
) ,1sinlimsinlimarcsinlim
1
000=
==
uu
uu
xx
uux xu arcsin= 0u .0x
) ,1tglimtglimarctglim
1
000=
==
uu
uu
xx
uux xu arctg= 0u .0x
. , , 1, , , ),(tux = 0)(lim
0=
tu
tt (
a), ).)(lim0
=
tutt
2. :a) ;4sin
5tg3sinlim0 x
xxx
) ;1lim 2
2sin3
0
2
xe x
x
) .2coscos
32lim22 23
0 xxxx
x
: , 1
, :
a) ;21
141513
44sin4
55tg53
3sin3lim4sin
5tg3sinlim00
=
=
=
xx
xx
xx
xxx
xx
) ;121ln1222sin
2sin31lim121lim 2
2
2
2sin3
02
2sin3
0
22
==
=
ex
xx
ex
e xx
x
x
) )cos1()2cos1()13()12(lim2coscos
32lim23
0
23
0
2222
=
=
xxxx
xx
x
xx
x
.98ln3
2
21
214
3ln22ln3cos1
)2(2cos14
2132
3123
lim22
2
2
2
3
0
22
=
=
=
xx
xx
xx
xx
x
-
2
58
3. :a) ;
2)3ln(coslim 320 xx
xx +
) .32235lim 2
3
1++
xxx
x
:a)
2))13(cos1ln(lim
2)3ln(coslim 320320 =+
+=
+ xxx
xxx
xx
;49
029
2112
9)3(
13cos13cos
))13(cos1ln(lim 20 =+
=
+
+=
xxx
xx
x
) : .1= xu 0u 1x
4238lim
3)1(2)1(2)1(35
lim32235lim 2
3
02
3
02
3
1=
++
=
+++
++=
++
uuu
uuu
xxx
uux
.161
483
3124
83
83
1831
lim2
31
0==
+
+=
uu
u
u
4. :a) ;32
12lim1 x
x xx
+ ) .2sin1lim
1
0
x
x
x
+
: a) ).
a) =
+
+=
+
+=
+
111
3241lim132
121lim3212lim
x
x
x
x
x
x xxx
xx
=
++=
+
+
)1(32
4
432
432
11lim
xxx
x x ,11lim 2
32
114lim32
)1(4lim
eeux
xxx
u
u
xx
==
+ +
+
4
32
+=
xu u ;x
) ,)1(lim2sin1lim2sin1lim1
21
2
2sin
21lim1
0
2sin
2sin
1
0
1
0
0
eeuxxx
x
uu
x
x
x
x
x
x
x
==
+=
+=
+
2sinxu = 0u .0x
-
59
1. :a) ;
)2)(3(3)12)(13(lim 2xxx
xxx ++
+
) ;)21)(1()31)(21lim xx
xxx
+++(
) ;
11619lim 2
2
++
xx
x
) ;10
1)31)(21(lim 20 xxxx
x +
++
) ;1)71)(51(1)31)(1(lim
0+
xxxx
x) ;
16)3(9)2(lim 2
2
1+
+ x
xx
) ;1)3(8)2(lim 3
3
4
xx
x) ;
631)1(lim 42
3
0 xxxx
x +
+
) ;)4)(8()1()2(lim 23
2
2
+ xx
xxx
) ;62
103lim 22
2
+ xx
xxx
) ;11lim 21
xx
x) ;
11lim 3
3
1 ++
xx
x
) ;34122lim
2+
+ x
xx
) ;11lim
3
1
xx
x) ;
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3
0
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xx
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xxx
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x x
+
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1
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xx
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22
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xx
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3
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50
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xx
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60
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6
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6
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332
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1
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121
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00 .
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1. =
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xx
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+
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+
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00
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,
.
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xxx
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4
xxxx
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2
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xxx
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xx
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0 xx
xx
x
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62
xxxx
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+
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62
0 xxxx
x +
+
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2
0 xxxx
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52
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x
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+
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52
x
x
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+
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31
1lim 31
+
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22
+
++
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xx
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12lim2x
x xx
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+
-
2
64
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25 x
x xx
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41lim1
0
x
x xx
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0
xxx
x
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2
2
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23
1 ++
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121
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xxx
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n
n
xx
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2
1
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nxxx nx
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1
1
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xx
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6+
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xxx
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0
3 x
x xx
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1
0
x
x xx
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xx +
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143
0
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143 3 xx
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x
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x
x
x
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xxmx
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2
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22
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xxx
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3
70
. 0x , , . ,
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f .0x . , .
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71
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2
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===
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121lim)(lim)01(
0101=
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++
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3
72
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2 xx
xxxf
: f , 1x .
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1=x .
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75
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33
2
xaxxaxxxf .Ra
, f R.9. :: RR f
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+
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x
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3
76
2 , ,
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f 0x .
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xx=
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77
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00
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xx
xx xexf =)(
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x
x
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Ex= )(inf xfm
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)(afM = ( f -).2. R),(: baf ),,( ba ),(lim
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),(lim0
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xfmbx
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xfMax +
=
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fmEx
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1)( nna 1)( nnb ( ...)......,... 2121 nn bbbbaaaa .0)(lim =
nnn
ab ( 1, 3.1), ,limlim cba nnnn ==
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82
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84
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-
85
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86
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>=
-
87
1.
f
g
.
),
(R
f
,
,
+gf
gf
gf
)0)
((
x
g
.2.
-
.3.
-
.
)
(:
RR
EE
f
-
Ex
0
,
:
1.
).(
)0(
)0(
00
0x
fx
fx
f=
=
+
2.
0
>
0>
,
E
x
a .1a f - R .,ln)( R= xaaa xx
: ,ln)( aaa xx = ,0>a ,1a .Rx (8)
. (8), .,ln)( R== xeeee xxx
: .,)( R= xee xx )8(
: a) ;2ln2)2( xx = ) .3,0ln)3,0())3,0(( xx =
3.7. 11. ,),0(: R+f .ln)( xxf = f ),0( + ).,0(,1)(ln += xxx
-
4
104
: ,1)(ln xx = ).,0( +x (9)
. 11.
12. ,),0(: R+f ,log)( xxf a= ,0>a .1a f - ),0( + ,ln
1)(log axxa = ,0>a ,1a ).,0( +x
: ).,0(,1,0,ln1)(log +>
= xaaaxxa (10)
. 12.. ,
axxa ln
lnlog = - (9).
: a) ;2ln1)(log2 xx =
) .10ln1)(lg xx =
1. f :fD a) ,: RR f ;)( 8xxf = ) ,: RR +f ;)( 7= xxf) ,: RR +f ;)( 4 xxf = ) ),,0(: +Rf ;3)( xxf =) ),,0(: +Rf ;2
1)(x
xf
= ) ,),0(: R+f ;log)( 3 xxf =
) ,),0(: R+f ;log)(31 xxf = ) ,: RR f .)( 5 xxf =
2. RDf : :0xa) ,log)( 7 xxf = ;70 =x ) ,lg)( xxf = ;10
10 =x ) ,)( 2xxf = ;600 =x
) ,)( xxf = ;490 =x ) ,2)( xxf = ;50 =x ) ,25)( =xf .640 =x3. RDf : :0x
a) ,)( 3 xxf = ;10 =x ) ,2)( xxf = ;00 =x) ,log)( 8 xxf = ;20 =x ) ,)( 5xxf = .10 =x
A
4. f :fD a) ,),0[: R+f ;)( xxxf = ) ,: RR f .)( 5 23 xxxf =
5. f ,fD :: RDfa) ;)( 7 xxf = ) ;||)( xxf = ) );(log)( 24,0 xxf = ) .2)( ||xxf =
-
105
4 4.1. , . ,:, RR gf ,)( 3xxf = .,)( R= cexg x :a) ;)( + gf ) ;)( fc ) ;)( gf
) ;)( gf ) ;
gf ) .))(( gf o
. 13. RIgf :, ( )RI
,0 Ix gf + 0x ).()()()( 000 xgxfxgf +=+
))(())((lim)()( 00
00 xxgfxxgfxgf
x=
+++=+
).()()()(lim)()(lim 0000000
0xgxfx
xgxxgx
xfxxfxx
+=+
++
=
. f g I, gf + gfgf +=+ )( . (1) ,)()()(,: 3 xexxgxfxhh +=+= RR (1),
:.3)()()( 233 xxx exexex +=+=+
6. :: RDfa) |,cos|)( xxf = ;20
=x ) |,2|)( xxf = ;00 =x
)
>
= ,0,2,0,3)(
xx
xxxf .00 =x
7. RDf : :0xa) ,7)( 2xxf = ;30 =x ) ,sin)( xxf = ;30
=x
) ),(log)( 327 xxf = ;270 =x ) ,5,2)( xxf = .10 =x
8. ,: RR f
-
4
106
. , nfff +++ ...21 I
==
=
n
kk
n
kk ff
11
.)( (1)
. )1( . 14. RIf : ( )RI Ix 0 ,Rc fc 0x ).()()( 00 xfcxfc =
. 14.. 1. f I ,Rc -
fc fcfc = )( . (2) ,3)(,: xexhh = RR .3)(3)3( xxx eee ==
2. 1=c .)( ff =3. f , g I, gf
gfgf = )( . (3)
,)(,: 3 xexxhh = RR :
.3)()()( 233 xxx exexex ==
15. RIgf :, ( )RI ,0 Ix R Igf : 0x
).()()()()()( 00000 xgxfxgxfxgf +=
.0 Ix g ,0x ,0x
).()(lim 00
xgxgxx
=
=++
= x
xgxfxxgxxfxgfx
)()()()(lim)()( 000000
=+++++
= x
xgxfxxgxfxxgxfxxgxxfx
)()()()()()()()(lim 000000000
)()()()()()(lim 0000000 xxgxxgxfxxgx
xfxxfx
=
+++
+=
).()()()( 0000 xgxfxgxf +=
-
107
. f g I, - gf
gfgfgf += )( . (4)
,)()()(,: 3 xexxgxfxhh ==RR :
).3(3)()()( 23333 xexexexexexex xxxxxx +=+=+= 2
. , nfff ...21 n I -
.............)...( 21212121 nnnn ffffffffffff +++=
16. RIgf :, ( )RI Ix 0 ,0)( 0 xg g
f 0x
.)(
)()()()()(0
20000
0 xgxgxfxgxfxg
f =
g ,0)( 0 xg ),( 0xV
0)( xg ).( 0xVx x , ).( 00 xVxx +
=+++
=
+
=
xxgxxg
xxgxfxgxxfx
xgfxxg
f
xgf
xx )()()()()()(lim
)()(lim)(
00
0000
0
00
00
=
+
+
+= xxgxxgxfxgx
xfxxfxgxxg xx
)()()()()()(lim)()(1lim 0000000000
)()()()()())()()()((
)(1
02
00000000
02 xg
xgxfxgxfxgxfxgxfxg
==
),()(lim( 000 xgxxgx =+ g ).0x
. 1. f , g I 0)( xg ,Ix g
f
2ggfgf
gf
=
. (5)
2. 1=f (5) : 2
1gg
g
=
. (6)
-
4
108
4.2.
17. .tg)(,|2\: xxfkkf =
+ RZR f -
+ ZR kk |2\ .2\,cos
1)( 2
+= ZR kkxx
xf
cossincos
cos)(cossincos)(sin
cossin)tg()( 2
22
2 =+
=
=
==
xxx
xxxxx
xxxxf
.|2\,cos1
2
+= ZR kkx
x
: .|2\,cos1)tg( 2
+= ZR kkxx
x (7)
18. .ctg)(,}|{\: xxfkkf = RZR f - }|{\ ZR kk }.|{\,
sin1)( 2 ZR = kkxx
xf
: }.|{\,sin
1)ctg( 2 ZR = kkxxx (8)
. 18.
.)(,:
3
xexxhDh = R
:
.)3()3(3)()()(2
2
2
2
32
2
333
xx
x
x
xx
x
xx
x exx
exex
eexex
eexex
exxh ===
=
=
4.3. 19. ,: 21 IIf R2: Ig , 21, II . f ,10 Ix g
,)( 200 Ixfy = R= 1: Ifgh o 10 Ix ).())(()( 000 xfxfgxh =
:Ixxfxfgxfg = ),())(()))((( . (9)
-
109
4.4. 20. JIf : ( ), RJI . f Ix 0 ,0)( 0 xf IJf :1 , )(IfJ = , )( 00 xfy =
.)(1)()(
00
1
xfyf =
. JIf : I, ,0)(xf .Ix , IJf :1 J
,,)(1))(( 1 Jyxfyf =
.)(xfy = (11)
1.
.sin)(,]1,1[2,2: xxff =
.)( 1 f:
2,20x 0cos)()(sin 00 = xx
20. , arcsin1 =f ).1,1(0 y .sin 00 xy = .arcsin 00 xy = (11) :
).1,1(,1
1sin11
cos1)()(arcsin 02
002
00
=
== yyxx
y
, :.)1,1(,
11)(arcsin
2
= xx
x (12)
. ,: 21 IIf ,: 32 IIg R3: Ih , R1:)( Ixp , ))(()( xfghxp oo= 1Ix ( ) )())(())(()( xfxfgxfghxp = . (10)
:
a) ;2)(,: 3xxhDh = R) .2coslog)(,: 2 xxpDp = R:a) .28ln)3(2ln2)2( 333 xxx x ==
) 2)2sin(2ln2cos1)2()2(sco)2(cosglo)2cos(log 22 xxxxxx ==
=
.ln22tg2
2ln2cos2sin2 x
xx
==
-
4
110
2. .cos)(],1,1[],0[: xxff = .)( 1 f:
,2arcsinarccos
=+ xx :
.)1,1(,1
1)(arccos2
= xx
x (13)
3. ,2,2:f
R . tg)( xxf = .)( 1 f:
2,20x 0
cos1)( )(tg
020 = x
x 20 arctg1 =f ,0 Ry . tg 00 xy =
.1
1tg11cos
cos11
)(1)()(arctg 2
0020
2
02
00 yx
x
xxfy +
=
+===
=
, :
R+
= xx
x ,1
1)arctg( 2 . (14)
4. .ctg)(,),0(: xxff = R .)( 1 f:
,2 arctgarcctg
=+ xx :.,
11) arcctg( 2 R+
= xx
x (15)
4.5. f(x) = u(x)v(x), u(x) > 0 ,: RIf ,)()( )(xvxuxf = .,,0)( R> IIxxu
, , , (3), (8) 3. :
.)()( )(ln)()(ln)()( xuxvxuxv eexuxf
xv===
)(ln)()(,: xuxvexfDf =R . :.))((ln)())(ln)(()())(ln)(()()( )()(ln)()(ln)( ==== xfxfxuxvxuxuxveexf xvxuxvxuxv (16)
,)()( )( xvxuxf = :0)( >xu
.ln)(
+= uuvuvuu vv (17)
-
111
4.6. RIf : I . )(xf
, , x, f , , x. , .f
,: RR f .)( 2 xexxf =
:.2)()( 22 xxx exxeexxf +==
).24(222)2())(( 222 ++=+++=+= xxeexxexeeexxexf xxxxxxx
. RIf : . , f - ,Ix 0 f 0x f .0x f 0x
( ) f x0 ).( 0xf
, .)()(lim)()()( 00000 x
xfxxfxfxfx
+
==
. f I, , f I.1. ,: RR f ,53)( 23 += xxxf : ,63)( 2 = xxxf
.66)( = xxf2. ,: RR g ,cos)( xxg = : .cos)(,sin)( xxgxxg ==
:
a) ,: ** ++ RRf ;)(xxxf = ) ,: * RR +f xxxf x 5lg)( = (.
3).:a) (16) .))((ln)()( = xfxfxf, ).1(ln)ln()(ln)( +=== xxxxxxxx xxxxx
) .)5(lg5lg)()5lg()( +== xxxxxxxf xxx .)(,: xxxgDg = R
(17) uuvuvuvuv +== ln)ln()(ln
,)(1)(ln = xx
x xx
x .2
)ln2()(x
xxxx
x +=
.lg5lg)ln2(5,010ln25lg)ln2()( 15,0 exxxxx
xx
xxxxf xxxx
++=++
=
-
4
112
( -) f .0x : )( 0xf .
n- , ,2,* nn N f .0x : ).()()( 0)1(0)( xfxf nn = )()( xf n .d
dn
n
xf
. 1. , , , ( , ).2. , f - f , .)0( ff =1. ,: RR f ,sin)( xxf = : ,cos)( xxf = ,sin)( xxf =
,cos)( xxf = .sin)( xxf IV =
: ,2sin)(sin)(
+=nxx n .Nn
2. : ,2cos)(cos)(
+=nxx n .Nn
3. ,: RR g ,)( xexg = : ,)( xexg = ,)( xexg = .)( xexg =
: ,)( )( xnx ee = .Nn
, .: xa + n, *Nn , :
,,...)( 332
210 R+++++=+ xxAxAxAxAAxan
nn (18)
nAAAAA ...,,,,, 3210 , . ,0A 0=x (18) :
.0naA = (19)
1A (18) :)...())(( 33
2210 +++++=+
nn
n xAxAxAxAAxa , ....32)( 12321
1 ++++=+ nnn xnAxAxAAxan (20)
0=x (20), 11 Aan n = . .1
1
1
=
nanA
(20) 0=x , 2A (!):
.21)1(
2)1( 22
2
=
=nn annannA
, : ....,,, 43 nAAA -
-
113
1. f :: RDfa) ;5)( 6xxf = ) ;)( xexf = ) ;log5,0)(
31 xxf =
) ;5)( 23 xxxf = ) ;237)( 2 += xxxf ) .0102log2)( 5 += xxf2. ,fD f -
fD :: RfDfa) ;)( xxxf += ) ;log)( 53 xxxf += ) ;)( xxexf = ) ;ln)( xxxf =) ;log)(
51
3 xxxf = ) ;11)(
2
+= x
xxf ) ;2
)( 3 xxxxf+
= ) ;ln)( xxxf =
) ;3)( = xexf
x
) ;2)( 2 xxxf = ) .2log4)( 2 xxf =3. f ,0x :
a) ,: RR f ,1)( 2xxxf = ;20 =x ) ,),0(: R+f ,2log)( 5 xxxf = .25,00 =x
4. ,7331)( 23 tttts ++= s ,
, t , . :a) ;) 2=t ;) .
5. , - ttts 46)( 21 += ,63)( 232 tttts ++= s , - , t , .a) , .) .) .) , .
A
k ),( * nkk N (18) 0=x - , : kkn Akaknnnn =+ ...21)1(...)2)(1( .
....21)1(...)2)(1( kn
k akknnnnA
+=
, kknnnn
+...21
)1(...)2)(1(
, .knC, ,knknk aCA = .)!(!
!...21
)1(...)2)(1(knk
nk
knnnnCkn
=
+=
.......)( 22211 nkknknn
nn
nnn xxaCxaCxaCaxa ++++++=+ (21)
, , (21).
-
4
114
6. f :: RDf
a) ;cos)( 25 xxxxf += ) ;logsin)( 53,0 xxxxf += ) ;53ln5)( 2xxxxf +=) ;3
172)( 96 += xexxf x ) ;ln4sin7cos5)( xxxxf = ) ;sin5)( 4 xxxf =) ;ln8)( 3 xxxf = ) ;log6,06)( 35 xxxf = ) );3ln(5)( 2 xxxf =) ;tglog2)( 35 xxf = ) ;4sin6)( 23 xxf x= ) .ln
)13cos()( 22
xxxf =
7. RDf : :0xa) ,2cos)( 2 xxf = ;30
=x ) ),13(lg)( 2 = xxf ;3
20 =x ) ,)( 2+= xxxf .10 =x
8. ,: RR f
+++
=
.),0[,),0,(,)(
2
2
xcbxaxxexf
x
-
a, b c, f .00 =x
9. ,: RR f :a) ;cos2)( xxf = ) ;2)( 2xexf = ) .2sin2)( xxf =
10. R ,0)( = xf :a) ;2sin2)( 2 xxxf += ) .32cos)( xxxf =
11. R ,0)( > xf :a) ;36)( 23 xxxxf += ) ).6cos(3)( += xxxf
12. f :: RDfa) ;652)( 23 = xxxf ) ;3sin2)( xxf = ) ;5)( 2xexf =) ;3)( 2xxf = ) ;ln)( xxf = ) ;3arccos)(
xxf =
) ;11)(
+
= xxxf ) ;
)1(3)( 2
2
=
xxf
x
) .)()( 1= xxxf
13. )(3)(5)( xfxfxf + , ,2,2:
Rf .arctg)( xxf =
14. .)( tts = , - .
15. F, m, ,4)( 23 ttts = m , s t .
16. ),0()5(f .)( 32 xexf x =17. , ,...32 321 nnnnn nCCCC ++++ },...,,2,1{, nkCkn
.18. :
....)( )()0()1()1(1)1()1(1)0()(0)( nnnnn
nn
nn
nn gfCgfCgfCgfCgf ++++=
-
115
5 )(: RR IIf I Ix 0 .
, , .)()(lim)( 00
00 xxfxxfxf
x +
=
(1)
(1) , ),()()()( 000 xxfx
xfxxf +=+
(2)
.0)(lim0
=
xx
(2) , xxxxfxfxxf +=+ )()()()( 000 ,
.)()()( 00 xxxxfxf += (3)
(3) , )( 0xf f 0x : ,)( 0 xxf - , ,)( xx 0)( x .0x
. ,)()(,: 0 xxfxgg = RR f 0x ).(d 0xf, xxfxf = )()(d 00 . (4)
.)(,: xxff = RR
: ,1)( 0 = xf .d xx = (4), .d)()(d 00 xxfxf =. f I, :
,d)()(d xxfxf = .Ix (5)
1. ],1,1[: Rf ,sin)( xxf = :
.dcosd)(sin)(sind)(d xxxxxxf ===
2. ,),0(: R+g ,log)( 8 xxg = .8lndd8ln
1)(d xxxxxf ==
f , 0x , 4.11. ))(,( 00 xfxA Gf . ,ABx = AB
BCxf == )(tg 0 (. ,ABC ).90)(m =B ABxfBC = )( 0 , ).(d)( 00 xfxxfBC ==
-
4
116
f 0x : )( 0xf f
))(,( 00 xfx , x , )(d 0xf ))(,( 00 xfx Gf , x - f (. 4.11).
(3) (4) :
)(d)()( 000 xfxfxxf + , (6) .BCBD
(6) :.)()()( 000 xxfxfxxf ++ (7)
(. . ) x .)( 0 yxxf + , A, - f , A Gf .
(7) .
,: RR f 154)( 2 = xxxf , - .1,1=x
:.1,011,1 0 xxx +=+== ,10 =x .1,0=x f(1) :)1(f ;1215114)1( 2 ==f 18)( = xxf .7118)1( ==f
(7), .3,111,07121,0)1()1()1,1( =+=+ fff f 1,1=x : .26,11)1,1( =f
. (7), :1. .2
111 xx ++ (8)
2. .,1)1( ++ Nnxnx n (9)
. (8) (9).
: a) ;008,4 ) .)003,1( 100:a) (8), :
.002,2001,12002,02112002,012)002,01(4008,4 ==
++=+=
, : .00199,2008,4
y
. 4.11O x
AB
C
)( 0 xxf +
)( 0xf
xxx +00x
)( 0xf
fG
)(d 0xf
D
-
117
) (9) : .3,1003,01001)003,01()003,1( 100100 =++= , : .3493,1)003,1( 100 (7)
0x , . , .
. (7)(9) .x
.31sin : .sin)( xxf = .1806sin)130sin(31sin
+=+=
, .18061806sin
+=
+ f ,60 =x .180=x .180661806
+
+ fff ,2
16sin)( 0 ==xf ,2
36cos)( 0 ==
xf :
.52,018023
21
18023
2131sin
+=+
, :
,0)(d =c ;Rc ,d)(d 1 xxx = ;R ;2d)(d
xxx =
;d)(d xee xx = ;dln)(d xaaa xx = ;d)(lnd xxx =
;d1d 2xx
x =
;lnd)(logd ax xxa = ;dcos)(sind xxx =;dsin)(cosd xxx = ;
cosd) tg(d 2 x
xx = ;sin
d) ctg(d 2 xxx =
;1d)(arcsind
2xxx
= ;1d)(arccosd
2xxx
=
;1
d)arctg(d 2xxx
+= .
1d)arcctg(d 2x
xx+
=
(. 4) - .
1. .d)3(d3d)(d 2233 xxexxexxxexe xxxx +=+=2. .d3cos3)3(sind xxx =
-
4
118
!
1. , (6), :: RR fa) ,2)( 3 xxxf = ,04,11 =x ;98,02 =x ) ,15)( 2 += xxxf ,04,251 =x .98,02 =x
2. , (7), (8) (9):a) ;)0008,1( 200 ) ;)996,0( 7 ) ;011,36 ) ;998,0 ) ).05,1ln(
3. :: RDfa) ;2)( 3 xxxf += ) ;1)( x
xxf
= ) );1sin()( += xxf ) ;2)( 3xxf = ) .2cos)( xxf =4. :: RDf
a) ;log)( 2 xxxf = ) ;)( 42 xexxf =) ;5ln)5(ctg)( xxxxf += ) .53ln3)( +=
xxf
5. :: RDfa) ,5)( 2 += xxf ;20 =x ) ,cossin)( xxxf = ;30
=x
) ),3(log)( 22 += xxf .10 =x6. :: RDf
a) ;5)( 74 += xxxxf ) ;7)1ln(32)( 2 += xxf x ) .tg)( 5 22 xxxxf =7. :: RDf
a) ,5cos2sin)( 3 += xxxf ;60
=x ) ,arccos53arctg)( xxxf += ;10 =x) ,3arcsin75)(
2 xxf x += ;00 =x ) ,)( 23 xexxf = .20 =x8. (7), :
a) ;46cos ) ;2,11lg ) ;93,0e ) ;004,06sin
) .004,1
120
6 .
.6.1.
( ) .
21 ( 1). : RIf - I .0 Ix 0x f, .0)( 0 = xf
1 (16011665) .
-
119
, 0x f . -
)( 0xV ))(( 00 IxVx , ).(),()( 00 xVxxfxf ),( 0xVx ,0xx < ,0
)()(0
0
xxxfxf
,),( 00 xxxVx > .0)()(
0
0
xxxfxf
, f 0x , , )()( 00 xfxf ==),( 0 xf = .0
)()(lim)(,0)()(lim)(0
0
00
0
0
00
00
=
=
>
< xx
xfxfxfxxxfxfxf
xxx
xxx
, 0)( 0 xf ,0)( 0 xf , .0)( 0 = xf , 0x -
f . , f f.
. - , - f ))(,( 00 xfx Ox(. 4.12).
. f 0x . , 0x , , f ., ,)(,: 3xxff =RR -
,00 =x 00 =x - f (. 4.13).
, .
y
. 4.12O x0x
fG
1x 2x 3x
)( 0xf
y
. 4.13
O x
3xy =
11
11
6.2. , ,
. 22 ( 1). ],[: baf R1) ],,[ ba2) ),( ba 3) ),()( bfaf = ),( bac , .0)( = cf
1 (16521719) .
-
4
120
f ],,[ ba
( 3, 3.1), .
.,),(sup),(inf],[],[
R==
MmxfMxfmbaxbax
: .; MmMm
-
121
1. -
(.4.15).
2. - - (. 4.16).
y
. 4.15O x
fGa bc
y
. 4.16
O x1c
fG
2ca ba)
y
O1c
fG
2ca
b
)x
6.3. 23 ( 1). .],[: Rbaf f ],[ ba - ),,( ba ),( bac , ).()()()( abcfafbf = ,],[: RbaF
.,)()( R= mmxxfxF F - ],[ ba ).,( ba Rm , ),()( bFaF = .)()( ab
afbfm
= F , ),( bac ,
.0)( = cF mxfxF = )()( 0)( = cF , .)( mcf =
, abafbfcf
=)()()( , )()()()( abcfafbf = (1).
. f ).,( bax - , ))(,( afaA ))(,( bfbB , ,)()( 1mab
afbf=
- f
))(,( cfc .)( 2mcf = 21 mm = , .
, , , ,fG AB (. 4.17).1 (17361813) .
c
. 4.17
y
O x
fG
a b
a)
AB)(bf
)(af
y
O x
fG
)
A
B
a b1c 2c
)(cf
-
4
122
=
]2,1(,4],1,0[,26
)(,]2,0[:2
xx
xxxff R
c.: f )1,0[
].2,1( ,4)01()1()01( =+== fff f 10 =x, , ].2,0[
= ].2,1(,4
),1,0[,4)(
2 xx
xxxf
, .4)1()1( == ff .4)1( =f , f ).2,0( )2,0(c , ),02()()0()2( = cfff
.2)( = cf f , 24 = c )1,0(c 242 = c ),2,1(c 5,01 =c 22 =c . , : 1c .2c
: .2;5,0 21 == cc. 1. (1) , .2. , c .3. ., ),()( bfaf = (1) , ,0)( = cf .4. - 5 ( 2, 1, 1.1). 1. RIf : ,,0)( Ixxf = f
I.2. RIgf :, I ,gf =
fg I.3. f V ,0x
}{\ 0xV .0x ,),(lim 00
R=
xfxx
)( 0xf .)( 0 = xf
. 3 - f .0x .
,
=
=
,0,0
,0,1sin)(,:
2
x
xxxxff RR -
,00 =x )(lim0 xfx .
-
123
6.4.
. , 1,
)()(lim
0 xgxf
xx, , 0)(lim)(lim
00==
xgxf
xxxx ,
.6.4.1. 0
0
24. I ,R)( I Ix 0 R}{\:, 0xIgf -. : 1) ,0)(lim)(lim
00==
xgxf
xxxx2) f g },{\ 0xI3) ,)(,0)( 0 IxVxxg I4) ( ) ,)(
)(lim0 xg
xfxx
)()(lim
0 xgxf
xx )(
)(lim)()(lim
00 xgxf
xgxf
xxxx
=
.
6.4.2.
25. I , Ix 0 R}{\:, 0xIgf . 1) ,)(lim)(lim
00==
xgxf
xxxx2) f g },{\ 0xI3) ,)(,0)( 0 IxVxxg I4) ( ) ,)(
)(lim0 xg
xfxx
)()(lim
0 xgxf
xx )(
)(lim)()(lim
00 xgxf
xgxf
xxxx
=
.
. 1. 24 25 .2. .x3. 24 25 0
0 .
4. )()(lim
0 xgxf
xx, )(
)(lim0 xg
xfxx
00
,,,, gfgf - , .)(
)(lim)()(lim
00 xgxf
xgxf
xxxx
=
, .1 (16611704) .
-
4
124
: ) ;2
3sinlim0 x
xx
) .2lim xx ex
+
:) .2
32
3cos3lim)2()3(sinlim0
02
3sinlim000
==
=
=
xx
xx
xxxx
) .02lim)()2(lim2lim ==
=
=
+++ xxxxxx eex
ex
. , , .6.4.3. 00 010 ,,,, 00 0,,1,,0
00
, 2.
1. : a) );ln(lim 20
xxx
+ ) ;1tg
1lim0
xxx ) ;lim0x
xx
+ ) .1
1lim2 x
x xx
+
+
:a) .0 .1
lnln2
2
x
xxx =
21)(,ln)( xxgxxf == , ,lnlim,),0(:, 0 =+ + xgf xR .1lim 20 +=+ xx
f g : 01)( = xxf ).,0(,02)( 3 += xx
xg
.02lim2
1lim)(
)(lim)()(lim)ln(lim
2
0
3
000
2
0=
=
=
==
+++++
x
x
xxgxf
xgxfxx
xxxxx
: .0)ln(lim 20
=+
xxx
) . , tg tg1
tg1
xxxx
xx
= .0
0
tgcos1coslim
cos tg
cos11
lim00
) tg() tg(lim)(1 tg
1lim 22
0
2
2
000=
+
=
+
=
=
==
xxxx
xxx
xxxxx
xx xxxx
.012cossincos2lim
2sin21
)1(coslim00
2sin21
1coslimsincos1coslim
0
2
0
2
0
2
0=
+
=
+
=
=
+
=
+
= x
xx
xx
x
xx
xxxx
xxxxx
: .01 tg1lim
0=
xxx
. ) , .0
0
-
125
) .00 .)( xxxf = ,ln)(ln xxxf = .)( ln xxexf =
,01
1lim
1)(lnlim1
lnlim)ln(lim2
0000=
=
=
==
++++
x
x
x
x
x
xxxxxxx
.1lim)(limlim 0)ln(lim
ln
000
0=====
+
+++eeexfx
xxxx
xx
x
x
x
: .1lim0
=+
x
xx
) .1
,11)(,:
2 x
xxxfDf
+
= R .11ln2)(ln
+
= xxxxf
, .421
12
lim00
21
11ln
lim)(lnlim2
2=
=
=
+
=+++
x
x
x
xx
xfxxx
: .11lim 4
2
=
+ ex