Matan 8dec

8/13/2019 Matan 8dec

1/50

()

7 2013 .

1


2/50

I 6

1 6

2 7

3 7

4 7

II 9

5 9

6 9

7 10

8 11

III 11

9 11

10 12

11 12

12 13

IV 13

13 13

14 15

15 15

V 16

16 16

17 16

18 16

2


3/50

19 17

VI 18

20 18

21 19

22 - 19

23 .. 21

VII 21

24 21

25 22

26 22

VIII , - 23

27 23

28 24

IX 24

29 2429.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2429.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

30 25

31 26

X , 27

32 27

33 28

34 28

XI 29

3


4/50

35 29

36 3036.1 . . . . . . . . . . . . . . . . . . . . . . . . . 31

37 32

37.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3237.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3337.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

XII 35

38 35

39 3639.1 . . . . . . . . . . . . . 36

39.2 . . . . . . . . . . . . . . . . . . . . . . . . . 36

40 36

41 37

XIII 37

42 3842.1 , . . . . . . . . . 38

42.2 . . . . . . . . . . . . . . . . . . 38

43 3843.1 0 . . . . . . . . . . . 3843.2 . . . 3843.3 . . . . . . . . . . . . . . . . . . . . 39

XIV 3943.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3943.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

XV 40

44 4044.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4044.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4144.3 . . . . . . . . . . . . . . . . . . . . . 41

45 41

4


5/50

XVI I 4245.1 , . . . . . . . . . . . . . . . . . . . . . . . . . . 44

XVII II 44

XVIII III 46

XIX 47

XX 49

46 49

47 49

5


6/50

I

1

- X Y

X Y ={(x, y)|xX, yY}

() R , :

1. RR +

R

(a) 0, x R x+ 0 = 0 +x= x(b)x R x: x+ (x) = (x) +x= 0(c) x,y,z R (x+y) +z=x+ (y+z)(d) x, y R x+y = y +x

1a, 1b, 1c G, G (). 1d, G ().

2. R

R

R

(a) 1, x R x 1 = 1 x= x.(b)x R \ {0} x1 :x x1 =x1 x= 1.(c) x,y,z R (x y) z= x (y z)(d) x, y R x y = y x

2a, 2b, 2c, G . 2d, .

1.+ 2. x,y,z R (x+y) z= xz+yz3.

x y R x y

(a)x R x x(b)x, y R, (x y) (y x), x= y .(c) : x,y,z R (x y) (y z)x z(d)x, y R (x y) (y x).

3a, 3b, 3c G, G -

. 3d, .

1.+ 3.x,y,z R, x yx+z y+z

6


7/50

2.+ 3.x, y: 0 x, 0 y0 x y P 1, 2, P (). 3, P .

4. ()

X R Y R :x, y: x

ycR :x

c

y., X= [0;

2), Y = (

2;2] c Q.

2 -

,

a0, a1, a2, . . . , an, . . .

,a0 ,a1a2 . . . an . . . - ( {0, 1, . . . 9}). , +0, 00 . . . 0, 00 . . . , a0, a1 . . . an999 . . . a0, a1 . . . (an+ 1)000 . . . (an= 9) ( ).

3

P Q , -

() f: PQ, , ..x, yPf(x+y) =f(x) +f(y)

f(xy) =f(x) f(y),

f . x y f(x) f(y), f .

3.1. .

4 , X Y,

X Y, ..xX yY, X Y y Y X.

.

f, x1, x2X :x1=x2f(x1)=f(x2)f, yYxX :y = f(x)

- . .

7


8/50

, X, X, . card X, |X|.

, . .

, X Y |X| |A|. [:|||||:]

X , N, .. card X =cardN =0 (-). X , , .

4.2.

R , -.

4.3 (). R , N.

. (0, 1). R, .. card(0, 1) = cardR. n :

0, a11a12 . . . a1n . . .

0, a21a22 . . . a2n . . .

. . . . . .

0, an1an2 . . . ann . . .

0, {a11, 0, 9}{a22, 0, 9} . . . {ann, 0, 9}. 0 1, n , .. 1- , 2- .. , , , .

[:|||||:]

4.4. ,

N.

8

9/50

, - (0, 1). a = a1a2a3 . . . an. a f(a) =

ni=1 ai 2i. -

, (0, 1) 2.

II

5

MR (), A(B)R :xM, x A(B x).M R , C R :xM, |x| C., A M( ), B M (

).aX () X, xX, x

a (x a). , () , , .

() - sup X. () infX.

= sup X xX, x

< xX :x >

= infX xX, x > xX :x <

6

6.1 ( ). , - .

. X R Y ={y R| x X, x y} ( ).X

= Y

= .

R :xX yYx y. X Y.

X Y Y = min Y= sup X. [:|||||:]

6.2 ( ). , - .

. X R Y ={y R| x X, y x} ( ).X= Y= .

R :xX yYy x. X Y. X Y Y = max Y= infX. [:|||||:]

9

10/50

X , sup X. sup X, .

: X = (0, 1), sup X = 1, infX = 0. , -, .

7 7.1 (). a n, 1 n , a.

1 + 1 + + 1 n

> a

7.2 (). a, n, n > a, ..a R n N :n > a.

. 7.1. [:|||||:]

7.3 ( 7.2). a b, 0 < a < b, k, (k 1)a b < ka.

.

(k 1)a b < kaka a b < kak 1 b

a< k

7.2 , n, b/a < n. k = min{n|n > b/a}. k

1 b/a. [:|||||:]

7.4. a b (a < b) . c, a b.

. a= 0, a1a2 . . . an . . . , b= 0, b1b2 . . . bn . . . . , a b . , , . i, ai < bi( , .. a < b). 2 :

1. bi> ai+1. c = 0, a1a2 . . . (ai+1)000 . . . . , , a b.

2. bi=ai+ 1. 2 :

(a) b = 0, b1b2 . . . bi . . . 0 . . . . j. c = 0, b1b2 . . . bi000 . . . 100 . . . , 1 j+ 1. a b . , - 1, a = 0, a1a2 . . . ai999 . . . , c= a.

(b) b = 0, b1b2 . . . bi000 . . . . , a= 0, a1a2 . . . ai999 . . . , .. - a = b. j > i, aj < 9. c = 0, a1a2 . . . ai . . . (aj + 1)1000 . . . . , b i- a j-. aj+1 = 1, ..

a= 0, a1a2 . . . ai999 . . . aj9999 . . . , c= a.

[:|||||:]

10

11/50

8

8.1. n, :

1. n= 1.

2. - k = n n= k + 1.

. . , n. m, :

1. n= m .

2. n, m, ( , m , ).

, m > 1, .. n = 1 ( 1). -

, m1 . , m1 , m . - 2. [:|||||:]

III

9

NX x1, x2, . . . , xn.N R .

(), MR (mR) :n N xn M(xn m). ..{xn} , AR :n N |xn| A.

..{xn} , {xn} . ..{xn} , AR n N :|xn|> A.

{xn} ,

AR N(A) :n N |xn|> A

.... , , , .

xn=n+ (1)n+1 n

2A0 R :Nn0 N |xn0|< A0 ( ....)

A0 = 1 :N N k0 N :|x2k+1|= 0< 1 =A0{xn} ,

>0

N() :

n N()

|xn

|<

11

12/50

10 -

{n} ....1.

{n

}.

. , > 0N() :n N |xn| < . , n N . n < N xn max{|1|, |2|, . . . , |N1}. = 1, A=max{|1|, . . . , |N1|, 1} n N, |n| A. [:|||||:]

2. {yn} , {n yn} .

.

>0N() :n N() |n|< A

A:n N yn A{n yn}: >0N() :n N |n yn|<

A A=

[:|||||:]

3. {n} .., {n n} {n n} .

.

>0N1() :n

N1 |n|0

N() :

n N

|xn

|<

1

|xn|>

1

=A

[:|||||:]

12

13/50

11.2. {xn} .. n N, xn= 0,

1xn

} .

.

A >0N(A) :n N |xn|> A 1|xn| < 1

A=

[:|||||:]

12

a R , U(a).U(a) = (a; a + ) -. U (a) = (a; a)(a; a+) =

0

U= U \ {a} . , ..{xn} , a,

limnxn=a U(a)N(U(a)) :n NxnU(a) (1)limn

xn=a >0N() :n N |xn a|< (2)

12.1. .

. 1- . xnU(a), a < xn< a + 0N() :n N |xn a|< a < xn < a+

L >0 :

n

N

|xn

| L= max

{|x1

|, . . . ,

|xN1

|,

|a+

|,

|a

|}

L xn, n, n < N. [:|||||:]

3. limn

xn = a, limn

yn = b, limn

(xn yn) = a b, limn

(xnyn) = ab,

limn

xnyn

= ab

, b= 0.

.

limn

xn=a, limn

yn=b

xn= a+n, yn= b+n,

{n

},

{n

} ..

xn yn = (a+n) (b+n) = (a b) + (n n) ..

xnyn= (a+n)(b+n) =ab+ (nb+an+nn) ..

{ xnyn

ab} , .

:

13.1.

limn

yn=b

= 0.

r >0,

N

N :

n N

|yn

|> r >0

.

>0N() :n N |yn b| 0N2 :n N2 |yn b|< 2

n N1 |xn a|< c a n N2 |yn b| yn, a bN :n Nxn yn, a bN :n Nxn> b, a bN :n Nxn b, a b

. 4 . , a < b. 14.1N :n N xn < yn ( -) xn 0N1 :n N1 |xn a|< a < xn< a+ >0N2 :n N2 |zn a|< a < zn < a+

a < xn yn zn< a+ limn

yn = a

[:|||||:]

15

16/50

V

16

n N xn xn+1(xn xn+1), .. () {xn} (). .

n N xn > xn+1 (xn < xn+1), .. () {xn} (). -.

17

17.1. .. ....,

.

. - - 2. .

{xn} , {xn} . sup{xn}= S >0N :S < xN

{xn} n N xn xNS < xN xn S < S+ lim

nxn = S= sup xn.

[:|||||:]

17.2. .. - ...., .

. - - 2. .

{xn} , {xn} . inf{xn}= S >0N :S+ > xN

{xn} n N xn xNS+ > xN xn S > S lim

nxn = S= infxn.

[:|||||:]

18

18.1. en=(

1 + 1n

)n, yn=

(1 + 1

n

)n+1. lim

nen = lim

nyn.

16

17/50

. yn.

yn1yn

=

1 +

1

n 1

n

1 +1

nn+1 =

n

n 1

n

n+ 1n n n

n+ 1=

n2

n2 1n

nn+ 1

(1 +)n 1 +n : n2

n2 1n

=

1 +

1

n2 1n

1 + n

n2 1 >1 +1

n=

n+ 1

n

n2

n2 1n

nn+ 1

>n+ 1

n n

n+ 1= 1

..n N yn1 > yn {yn}

1 +1

nn+1

= 1 +1

nn

1 +1

n> 2 1 +1

n= 2n+ 1

n > 2 ..{yn} .,{yn} -.

{en} ( ).

0< yn en =

1 +1

n

n1 +

1

n 1

0 n N

17

18/50

. 1..

{an} , an b1 {an} {bn} , bn a1 {bn}

- 2

bn

an

0

limn

bn = limn

an= c

an c bn n N ( inf{bn} sup{an})cn=1

[an; bn]

[:|||||:]

19.1 . ,

n=1

0;

1

n

= 0

VI

20

n1 < n2 < . . . . {xnk} -{xn}. ,{1, 3, 5, . . . } / - N, {3, 1, 5, . . . } /N, .. n1> n2. 20.1. {xn} , -, .

.

limn

xn=a >0N() :n N() |xn a|< nk Knk N |xnk a|< lim

nxnk =a

[:|||||:]

20.2. / .. .

.

AN(A) :n N |xn|> Ank Knk N |xnk |> A {xnk} ..

[:|||||:]

18

19/50

21

a ( ){xn}, -:

1. a {xn}

2. /{xnk} ..{xn}, a . 21.1. .

.

1 2 - a {xn}. - a, 1, 1

n, 2n

, . . . , 1n

, . . . . xk1 c -k1, xk2 k2, k2 > k1 .... , .. - a -{xn}. xk1 , xk2 , . . . , xkn, . . . , - a, ..|xkn a|< 1n .

2 1, {xn} , a. - a (, ). , - a {xn}.

[:|||||:]

21.2. - .. , ..

. .. , - 2 . 20.1.

[:|||||:]

22 -

22.1. .. /..{xn} . [a; b] :n N a xn b. [a; b] 2 . ( [a1; b1]) {xn}.

[a1; b1] {xn}. xn1 . [a1; b1] 2 . {xn}. [a2; b2]. xn2[a2; b2]. . xnk , k- , .. xnk[ak; bk].

{[ak; bk]} ..., 19.1!ck=1

[ak; bk]

limkak = limkbk =c limkxnk =c ( )[:|||||:]

19

20/50

() {xn} () -.

limn

xn= max{k|k }limn

xn= min{k|k }

22.2. .. .

. - ( - ).xn . m, M :n N m xn M.{x} ={x| x ..{xn}}. , -{x} x, x M. , -{x} ( c, c m). , inf{x}. , x= inf{x} {xn}.

1. , x . > 0 . x = inf{x} y : y < x y / {x}. , x {xn}.

x= inf{x} >0x : x x < x+

{x} x , {xn}. x , x {xn}, (x , x) ( (x ,x+)) {xn}, .. x .

2. , x > x {xn}. = (x x)/2, (x ,x + ) (x ,x + ) = . - x x+. , > 0 x+ {xn}. , - x , {xn}, , x {xn}.

[:|||||:]

22.3. ..{xn} limn

xn = x, limn

xn = x. > 0 (x ,x+) {xn}.

. , x+ 2

x 2

{xn}. , x + 2 {xn} - 22.2. - . [:|||||:]

{xn}- +()(), M >0N(M) :n Nxn > M(xn M). 22.4. .. - /, /, - .

. .{xn} . Knk : xnk > K. nk < nk+1, N = nk n N xn > K

{xn} . [:|||||:]

20

21/50

23 ..

23.1. .. - ...., -.

.

{xn} / - , .. lim

nxn = lim

nxn.

limn

xn = limn

xn = x. > 0 (x , x+) ... ,{xn} , ..{xn} -.

[:|||||:]

23.2. .. - ...., - /.

. {xkn}-{xk}. {xk}, .

{xk}. {xkn}. {xkn} {xk}. , , , {xkn}. [:|||||:]

VII

24

{xn} - ,

>0N() :n N, pN |xn+p xn|<

24.1. {xn} , N, - (xN , xN+) ..

>0N :n N, p N |xn xn+p|< |xN xN+p|<

xN < xN+p < xN+.. xN xN+ , ..

. [:|||||:]

24.2. -.

21

22/50

.

>0N() :n N, pN |xn+p xn|0 :n N |xn| A

A= max{|xN |, |xN+|, |x1|, . . . , |xN|}[:|||||:]

25

25.1. {xn} ...., -.

.

{xn} a: >0N() :n N()|xn a|<

2

p N |xn+p a|< 2

|xn xn+p|=|xn a+a xn+p| |xn a| + |xn+p a| {xn}

{xn} . {xn} .. 22.3 > 0 (x; x) (xn; xn+), , 0 x x


23/50

xn =n=1

an=a1+a2+a3

a2+a2+ a4+a5+a6+a7

a4+a4+a4+a4+ +a2n+ a2n+1+ +a2n+11+. . .

a2n+a2n++a2n

a1+ 2a2+ 4a4+ + 2na2n+ =n=0

2na2n

, xn xn+1, ..{xn} . , 17.1 .[:|||||:]

VIII

,

27

, S ={X} E, yE Xi. S E.

, S S, S S S .

, : S Xi. Xi , , , .. S

S Xi , , Xi.

27.1 (-). [a; b] - .

. : S [a; b] .

I1 = [a; b]. I1 . (.. +=).

I2. : , - I3. {In}, S. .. Ik+1 Ik, {In} (I1I2 I3 . . . ). In=

I12n1

0 {In} (...). 19.1 ...!cn=1 In. .. In [a; b], c [a; b].

S , c . (; ). = min{c ; c}. {In} 0, In, . In , , , , , In

S, , , (; ). , , , . [:|||||:]

23

24/50

28

x E, - E.

x E. , E = (; ), E = [; ]( ,

. , (x ; x + ) ).

28.1 (-). X .

. . .. X , [a; b] :x X x [a; b]. X x [a; b]U(x), X. ? ,, y : y / [a; b]. y < a, (y; a)

X. x[a; b]U(x), .

[a; b] (.. y[a; b] : y /U(x),

U(x) U(y), ). ,

[a; b] ( 27.1). X ( ), - , , , X , . [:|||||:]

x E , E, .. E

U(x) =

. ,

, , , .

IX

29

E R a R E ( a E). f:ER. ( R) .

29.1

limExa

f(x) =b >0()> 0 :x E, 0

25/50

: . , b - a.

, , x . :

limx f(x) =b >0x0:x E,|x|> x0 |f(x) b|<

29.2

limExa

f(x) =b {xn}:n N xn=a xn E, limn

xn=a {f(xn)} n

b

: x a, f(x) b.

, , , . , .

, ? f:NR. .. Ef = N. E. , = 0.3 () E. , , N ( =a, aN, ).

. , a , lim

Exaf(x). , lim

x0

sinxx

= 1 (,

), x= 0 .

30

30.1. .

.

.

V(b)U(a) : f(U(a)E) V(b) x U(a)E f(x) V(b) .

{xn} :xn = axnn

a. -

> 0N() N n N 0 0. > 0 , >0, xnU(a) |f(xn)b|< ,..{f(xn)}

nb, .

.

25

26/50

: limxa

f(x) = b , b= limxa

f(x) .

:

V(b)U(a) xU(a) :|f(x) b|

: 0, i

xi, |f(xi) b| 0. .. , {n}={ 1n}. , - xn, 0 0 :x U(a) |f(x) b| < . , U(a), x, b < f(x) < b+ , .. f(x). [:|||||:]

31

, >0 (a ; a) (a; a+) E.

b () f , :

>0 : ()> 0 : x E, a < x < a (a < x < a+) |f(x) b| a) xn E limn

xn=a {f(xn)} n

b

: limxa0

f(x) =b = f(a0) limxa+0

f(x) =b = f(a + 0).

, ,

26

27/50

x a , , . . a , , a. :

limxa

f(x) =b f(a 0) =f(a+ 0) =b(..

xE :a

< x < a

xE :a < x < a+

xE

0 0 = 1 :x :1 < x < 0 sgn x =1 | sgn x(1)| = 0 < x : 0 < x < 1sgn x = 1 | sgn x 1| = 0 < . lim

x00f(x) =1, lim

x0+0f(x) = 1.

.. 1=1, limx0 f(x).

X

,

32 , -

. :

32.1. limxa

f(x) =b, limxa

g(x) =c. :

1. limxa

(f(x) g(x)) =b c

2. limxa

(f(x) g(x)) =bc

3. limxa

f(x)g(x)

= bc, c= 0 xU(a) :g(x)= 0

. - ( - ). , {xn} :n N xn = a : lim

nxn =

a {f(xn)} n

b, {g(xn)} n

c. , .., ,

{f(xn) g(xn)} n

b c limxa

(f(x) g(x)) =b c. [:|||||:]

, , 32.1:

( , ), .

27

28/50

33

f(x) :X R, g(y) : Y R, x X : f(x) Y (f(X) Y). X g f :f g () g (f(x)). 33.1. f(x) :X R, g(y) :Y R, f(X) Y. lim

xx0f(x) =y0, lim

yy0g(y).

limxx0

g (f(x)) = limyy0

g(y).

. limyy0

g(y) = z0. U(z0)

V(y0) :y Y V (y0) g(y) U(z0). V(y0)W(x0) :x W(x0) f(x)Y V(y0). , :

U(z0)W(x0) :xW(x0) X f(x) Y V(y0)g (f(x))U(z0)

, . [:|||||:]

, - , , . -, , z0 - x0.

, . , , , , ,

. , limx0

(sinxx

)2. , (

) , :

f(x) = sinx

x g(x) =x2

. 33.1 f(x) g(x) f(x).

34

limxa

f(x) =b, limxa

g(x) =c b < c. :

1.U(a) :xU(a) :f(x)< g(x)

.

d: b < d < c. U1(a), U2(a) :xU1(a) |f(x) b|< 1 = d b xU2(a) |g(x) c|< 2=c d.

U(a) = U1(a) U2(a) :xU(a) |f(x) b|< d b |g(x) c|< c db d < f(x) b < d b d c < g(x) c < c df(x)< d= d c+c < g(x)

[:|||||:]

2. . f , g , h : E R x E f(x) g(x) h(x)

limxa

f(x) = limxa

h(x) =b.

limxa

g(x) =b.

28

29/50

.

>0 U1(a) :xU1(a) |f(x) b|< U2(a) :xU2(a) |h(x) b|< b < f(x)< b+ b < h(x)< b+

xU1(a) U2(a)

b < f(x)

g(x)

h(x)< b+ |g(x) b|< limxa g(x) =b.[:|||||:]

, . , : [N, ), U(a). 34.1. lim

xaf(x) =b, lim

xag(x) =c. , U(a), :

1. f(x)< g(x)b c2. f(x) g(x)b c3. f(x)< cb c4. f(x) cb c

XI

35

, f(x) , :

>0 >0 :x, x E : 0 0x, x E U(a) |f(x) b|< 2 ,|f(x) b|< 2 .|f(x) f(x)| |f(x) b| + |b f(x)|< .

, , .

f(x), {xn}, - a ( ), xn=a.

29


30/50

, {f(xn)} b. - {xn}. .. N()N :n N 0


31/50

f(x) g(x) ( ) xa, :U(a) :xU(a) E f(x) =(x) g(x), lim

xa(x) = 1

:f xa

g.

4.

f(x) = x6

1 +x4 x0

x6, .. x6

1 +x4 =

1

1 +x4 x6

f(x) = x6

1 +x4 x

x2, .. x6

1 +x4 =

x4

1 +x4 x2

(x) , , .

36.1

1. . f xa g, g xa f.

.

f= gg = 1 fg

xaf

(.. a

1, 1a

1). [:|||||:]

2. . f xa

g, g xa

h, f xa

h.

.

U1(a) :xU1(a) E f(x) =(x) g(x)U2(a) :xU2(a) E g(x) =(x) h(x)

U(a) = U1(a) U2(a) ( , .. - a) f(x) = (x) (x) h(x)f

xah, .. lim

xa((x) (x)) = lim

xa(x) lim

xa(x) = 1. [:|||||:]

3. U(a) :f(x)= 0, g(x)= 0 xU(a) E, f xa

glimxa

f(x)g(x)

= 1. ,

limxaf(x)

g(x) , f(x) = 0, g(x) = 0 (. ).

, f(x) g(x) xa, U(a) :xU(a) E f(x) =(x) g(x), xa, lim

xa(x) = 0

f(x) = o(g(x)), x a ( ) f(x) = o(g(x)), x a.:

1

x2 = o

1

x , x x2 = o(x), x

0

1

x2 =

1

x1

x x2 =x x

31

32/50

, 3, g(x)= 0 U(a), f= o(g)limxa

f(x)g(x)

= 0.

f(x) g(x) x a f = o(g), f(x) .

O o .

, , .

37

37.1

limx0

sin x

x = 1

. limx0+0

sinxx

limx00

sinxx

, ,

1.

A

Ox B

C

x(0; 2

). R. :

SAOB =1

2R2 sin x

SAOB () =1

2R2x

SOCB =1

2R2 tg x

,

0< sin x < x

33/50

limx00

sin x

x =

u=xx=u

u0 + 0

x0 0

= limu0+0

sin(u)u = l imu0+0

sin u

u = 1

[:|||||:]

37.2

limx0

(1 +x)1x =e

. , .

{xk} k

0 + 0 (xk 0, xk > 0) xk < 1. nk : nk 1xk < nk + 1 1nk+1 < xk 1nk . xk 0 + 0 nk + ( , ), ( , 1):

1 + 1

nk+ 1

nk


34/50

37.3

x0 x= 0sin xx sin x= x+ o(x)

1 cos x x22

cos x= 1 x22

+ o(x2)tg xx tg x= x+ o(x)

arcsin xx arcsin x= x+ o(x)arctg xx arctg x= x+ o(x)ex 1x ex = 1 +x+ o(x)

ln(1 +x)x ln(1 +x) =x+ o(x)(1 +x)m 1mx (1 +x)m = 1 +mx+ o(x)

:

1.

sin xxlimx0

sin x

x = 1

.

2.

(1 cos x)x2

2lim

x0

(1 cos x) 2x2

= limx0

2 2sin2x2

x2 = lim

x0

4sin2x

2

4

x

2

2= 1

3.

tg xxlimx0

tg x

x = lim

x0

sin x

x 1

cos x= 1

4.arcsin xxlim

x0

arcsin x

x = [y= arcsin x] = lim

y0

y

sin y = 1

5.

arctg xxlimx0

arctg x

x = [y= arctg x] = lim

y0

y

tg y = 1

6.

ex 1x limx0

ex 1x

= [t= ex 1] = limt0

t

ln(1 +t)= 1, ..:

limt0

ln(1 +t)t

= limt0

ln(1 +t)1t = ln lim

t0(1 +t)

1t = ln e= 1

( ) , .. - , .

7.

(1+x)m1mx limx0

(1 +x)m 1mx

= limx0

em ln(1+x) 1m ln(1 +x)

ln(1 +x) x

= limx0

em ln(1+x) 1m ln(1 +x)

ln(1 +x)x

= 1

34

35/50

XII

38

f:E R. f a E,

V(f(a))U(a) :xU(a) E f(x)V(f(a)), .. f(U(a) E)V(f(a))

:

>0()> 0 :x E,|x a|< |f(x) f(a)|<

, - , , -

. , - , - a a . :

f(x) =

{x, x[0; 1]3, x= 2

, E[f] = [0;1] {2}. , a = 2 E.

U(a) :

U(a)E ={a}, f(

U(a)E) = f(a) V(f(a)) V(f(a)). , f a= 2. , , a

-E. f a,

limxa

f(x) =f(a) limxa

f(x) =f(limxa

x)

, - a. a : f(x)C(a).

f X, -.

f(x) =x R, ..a R |f(x) f(a)|=|x a|< = . f(x) = sin x R, ..aR | sin xsin a|=|2sin xa

2 cos x+a

2 | 2| sin xa

2 |

2|xa2 |=|x a|< =.

f(x) a (), limxa+0

f(x)

limxa0

f(x)

= f(a). -

, f(x) [a; b], f [a; b](.. (a; b)) a b .

35

36/50

39

39.1

39.1. f(x) g(x) a, fg, fg, fg

(g= 0) a.

. (f(x) +g(x)). . lim

xaf(x) =f(a) lim

xag(x) =g(a). , -

, , limxa

(f(x) + g(x)) = f(a) + g(a), ,

(f(x) +g(x)) a. [:|||||:]

39.2

39.2. f: X Y, g : Y R, f(x) Y. f(x) C(x0), g(x) C(y0 =f(x0)), g(f(x))C(x0).

. limxx0

f(x) =f(x0) limxy0

g(x) =g(y0). , -

, , limxx0

g(f(x)) = limxy0

g(x) =g(y0) =

g(f(x0)), , g(f(x))C(x0). [:|||||:] , f(x) a, a,

.:

V(f(a)) :U(a)x(U(a) E) :f(x) /V(f(a)) -0>0 : >0xE, |x a|< , |f(x) f(a)| 0

f(x) = sgn x a= 0. limx00 =1=f(0) = 0= limx0+0 = 1. f(x) =| sgn x| a= 0, .. lim

x00f(x) = lim

x0+0f(x) = 1=f(0) = 0.

f(x) = sin 1x

a= 0, .. limx00

f(x), limx0+0

f(x), limx0

f(x).

, , -.

40 1. lim

xaf(x), f(x) a,

, a . , a , . ,

f(x) =| sgn x| f(x) ={

| sgn x|, x= 01, x= 0

, a= 0.

2. limxa0

f(x) =, limxa+0

f(x) =, =, a I . , a= 0 f(x) = sgn x.

3. limxa0

f(x), limxa+0

f(x),

a II .

36

37/50

, f(x) -[a; b], f(x) , x1, x2, . . . , xn, f(x) , lim

xa+0f(x), lim

xb0f(x).

41

f(x) x1, x2 E :x1 < x2f(x1)< f(x2). f(x) x1, x2 E :x1< x2f(x1) f(x2). f(x) x1, x2 E :x1 < x2f(x1)> f(x2). f(x) x1, x2 E :x1 < x2f(x1) f(x2).

41.1. f(x) [a; b], (a; b) - .

. , f(x) [a; b]. c[a; b).

{f(x)} f(x) x(c; b]. ,.. f(b) {f(x)}. ,{f(x)} , .. f(x) f(c)x(c; b] inf{f(x)}= . , = lim

xc+0f(x). = inf{f(x)}, >0

(0; b c) :f(c + )< + . .. f(x), x(c; c + ) f(x) +. , < f(x)< + |f(x) |< lim

xc+0f(x) =.

{f(x)} f(x) x[a; c). , .. f(a) {f(x)}. ,{f(x)} , .. f(x) f(c)x[a; c) sup{f(x)} = . , = limxc0 f(x). = sup{f(x)}, > 0 (0; ca) : f(c) > . .. f(x), x (c; c) f(x) . , < f(x) < + |f(x)| < limxc0

f(x) =. [:|||||:]

41.2. f(x) . I .

. , f(x) .f(x) c(a; b), x < cf(x) f(c)f(x 0) f(c). - =, - c,

38/50

42

42.1 ,

42.1. f:ER limxa

f(x) 0 : f(x) U(a)

E.

. limxa

f(x) =b >0()> 0 :xU(a) E |f(x) b| 0 (f(a) < 0). > 0 : f(x) >0 (f(x)< 0)

x

U(a)

E.

. f(x)C(a), >0() > 0 :xU(a) E |f(x) f(a)|0 f(a)> 0 f(a) + 0. A x[a; b] :f(x)< 0. , A= (.. aA) , ..

x

A

x < b.

sup A= p. , p

(a; b), .. f(a)< 0, f(b)> 0 42.3 1- a:x[a; a+1)f(x) < 0 2- b:x(b 2; b]f(x) > 0. , f(p) = 0. , 42.3 - p, , , .. sup A x (p;p] f(x) < 0 x (p;p+ ) f(x) 0. . [:|||||:]

43.2

43.2. fC[a; b], f(a) =, f(b) =. [min{; };max{; }]. p[a; b] :f(p) =.

38

39/50

. = , = , = = , p -. = . , < < . g(x) = f(x). g(x) C[a; b] (..g(a) = f(a) = < 0, g(b) = f(b) = > 0). p : g(p) = 0, ..f(p) = 0f(p) =. [:|||||:]

43.3

43.3. f(x) [a; b] ...., .

XIV

43.4

43.4. f(x)C[a; b], f .. . f(x) . n N xn[a; b] : f(xn) > n (.. ) {f(xn)} . ..{xn} [a; b], {xn} . , - /{xnk} : lim

kxnk =p. -

p[a; b]. .. f(x)C[a; b] {f(xnk)} k

f(p).

, /

{f(xnk)

}, , -

. -. [:|||||:]

, . , f(x) = 1x

- (0; 1), .

, M (m) () f(x) E, -:

1.x E f(x) M(f(x) m).2. >0x E :f(x)> M (f(x)< m+). , , f(x) () E,

() . : M = supE

f(x), m = infE

f(x).

, . :

f(x) =

{x2, x(0; 1)12

, x {0, 1}

sup f(x) = 1 ..x[0; 1] :f(x) = 1. .

39

40/50

43.5

43.5. f(x)C[a; b], (..x1, x2[a; b] :f(x1) = sup

Ef(x), f(x2) = inf

Ef(x)).

. f(x) [a; b],

. M, . . M, ..x [a; b] f(x) < M.

g(x) = 1Mf(x)

. x[a; b]Mf(x)> 0 (Mf(x))C[a; b]. - , g(x) [a; b]. -, A:x[a; b]g(x) A.Mf(x)> 0f(x) M 1

A,

, M . [:|||||:]

XV

44

44.1

f(x), (a; b). x (a; b) . x : x+ x (a; b). , x . y =f(x+x)f(x) f, .

, :

f(x)C(x)y = f(x+ x) f(x)x0

0

f x

f(x) = limx0

y

x= lim

x0

f(x+ x) f(x)x

, , f

(x0) = limx0xf(x0)f(x)

x0x . :1. f(x) =c = const.

f(x) = limx0

c cx

= 0.

2. f(x) =x.

f(x) = limx0

x+ x xx

= 1.

3. f(x) = sin x.

f(x) = limx0

sin(x+ x) sin xx

= limx0

2sinx

2 cosx+ x2

x = lim

x0cos

x+

x

2

= cos x.

40


41/50

4. f(x) = cos x.

f(x) = limx0

cos(x+ x) cos xx

= limx0

2sin

x+x

2

sin

x

2

x =

= limx0

sinx+x2 = sin x.44.2

() f(x) x ( ) yx

x0 0. : f(x) f(x 0) f+(x) f(x+ 0) . :

1. f(x), f(x), f+(x), f(x) =f(x) =f+(x).

2. f

(x), f

+(x) f

(x) =f

+(x), f

(x) =f

(x) =f

+(x).3. f(x)=f+(x), f(x).

, f(x) = |x|. f(0) = limx00

|x|x

=1, f+(0) =lim

x0+0

|x|x

= 1=1 f(0).

44.3

x x+ x

M

I

P

, tg = f(x+x)f(x)x

= arctg f(x+x)f(x)x

. 0 = limIM

.

, 0 x. 0 = arctg

limx0

f(x+x)f(x)x

=

arctg f(x)tg 0 = f(x).

45

f(x) x, f(x+ x)f(x) = y =Ax+ o(x), A x, o(x) =(x)x, limx0 (x) = 0.

41


42/50

45.1 ( ). f(x) x , f(x), A= f(x).

.

f(x) . x y = Ax+ o(x)

yx = A+

o(x)x = A+ o(1).

limx0

yx

=AA = f(x).

f(x) limx0

yx

= f(x). (x) = yx

f(x). limx0

(x) = 0, ..

(x) = o(1) yx

f(x) = o(1)y= f(x)x+ o(x).[:|||||:]

45.2 ( ). f(x) . x,

f(x)C(x).. y = f(x)x+ o(x)

x00 -

. [:|||||:]

, . , f(x) =|x| 0, - .

f(x) . x. y= f(x)x + o(x). , f(x)x . x dy = f(x)x. x:

x

, dx= x

.

x= (t), dx= (t)dt.

XVI

I

45.3 ( ). x = (t) . t0, y = f(x) . x0 = (t0), f((t)) . t0, f(t0) =f

(x)(t) = (f((t)))(t).

. y =f(x)x+ o(x) =f(x)x+(x)x. yt

=f(x)xt

+(x)xt

(). x = (t) , y

t t, (t).

, limt0

x = limt0

(t+ t) (t) = 0. x 0 (x) 0. () :

limt0

y

t

= limt0

f(x)x

t

=f(x)(t) =f((t))(t)

[:|||||:]

42


43/50

45.4 ( ). f:XY f1 :YX . f(x)C(x0), f1(y)C(y0 =f(x0)). f(x). x0 f

(x0)= 0, f1(y) . y0, (f1)(y0) = 1f(x0) .. x x0 X y y0 Y. , f1(y0) = x0, f1(y0 + y) =x0+ x, .. x= f

1(y0+ y)

f1(y0).

(f1)(y0) = limy0

f1(y0+ y) f1(y0)y

= limx0

x0+ x x0f(x0+ x) f(x0)=

= limx0

1

f(x0+ x) f(x0)x

= 1

f(x0)

[:|||||:]

45.5( ). f(x) g(x) . x, f

g, f

g, f

g (g

= 0) . x, (f

g) =

f g, (f g) =fg+f g,fg

= fgfgg2

.

. , f f(x),g g(x), h h(x).

1. h(x) =f(x) g(x).

h= h(x+ x) h(x) = (f(x+ x) g(x+ x)) (f(x) g(x)) == (f(x+ x) f(x)) (g(x+ x) g(x)) = f g

,h

x=

f

x g

x

x0 , f(x) g(x),

h(x) =f(x) g(x).2. h(x) =f(x) g(x).

h= h(x+ x) h(x) =f(x+ x)g(x+ x) f(x)g(x) == (f(x+ x)g(x+ x)

f(x+ x)g(x)) + (f(x+ x)g(x)

f(x)g(x))

h= f(x+ x)(g(x+ x) g(x)) +g(x)(f(x+ x) f(x)) =f(x+ x)g+g(x)f

,h

x=f(x+ x)

g

x+g(x)

f

x

x 0. f(x) x (.. . ) lim

x0

f(x+ x) =f(x). ,

h(x) =f(x)g(x) +g(x)f(x).

43

44/50

3. h(x) = f(x)g(x)

. g(x)= 0, g(x+ x)= 0 x.

h= h(x+ x) h(x) = f(x+ x)g(x+ x)

f(x)g(x)

=f(x+ x)g(x) g(x+ x)f(x)

g(x)g(x+ x) =

=

(f(x+ x)g(x)

f(x)g(x))

(g(x+ x)f(x)

f(x)g(x))

g(x)g(x+ x) =

=g(x)(f(x+ x) f(x)) f(x)(g(x+ x) g(x))

g(x)g(x+ x) =

g(x)f f(x)gg(x)g(x+ x)

h

x=

g(x)f

x f(x) g

xg(x)g(x+ x)

, ,

h(x) =g(x)f(x) f(x)g(x)

g2(x) .

[:|||||:]

45.1 ,

y x t, ..

{x= (t)

y= (t),

tT, , y=y(x) , t .

45.6 ( , ). . T, x0 , , ..t= 1(x). y(x) =

(t)(t)

.

. , y = (t) = (1(t)). : dx = (t)dt, dy = (t)dt.

dy=y (x)dx, dydx

= (t)

(t). [:|||||:]

XVII

II

, y=f(x) () c, U(c) :xU(c) x f(c)), x > cf(x) > f(c) (f(x) < f(c)). y =f(x) () c, U(c) :x U(c) f(x) < f(c) (f(x) >f(c)).

, ( ),

/ . y =f(x) , - .

44

45/50

45.7 ( / ). y = f(x). c f(c)> 0 (f(c)< 0), f(x) () c.

. f(c)> 0. .

f(c) = limxc

f(x) f(c)x c

>0

()> 0 :

x, 0 m, (a, b)

. f() = 0.

[:|||||:]

45

46/50

.. , 1 b a. , . :

f(x) = {x, 0 x 0 (f(x)< 0), f(x)

( ) (a, b).

. [x1, x2](a, b). - , f(x2)f(x1)

x2x1=f().

.. x2 > x1 f()> 0, f(x2)> f(x1). , f

()< 0, f(x2)< f(x1). [:|||||:]

45.12 ( ). , - f(x) = const [a, b] , f(x) = 0x[a; b].

46

47/50

. . -. [x1, x2][a, b]. ,

f(x2) f(x1)x2 x1 =f

() = 0f(x2) =f(x1) (.. x2=x1) f(x) =const

[:|||||:]

45.13 ( ). x(t) y(t):

1. x(t), y(t)C[, ].2. x(t), y(t) . (, ).

:

1.

(, ) :x()(y()

y()) =y ()(x()

x()).

2. x(t)= 0 t[, ], y()y()x()x()

= y()x()

.

.

1. F(t) =x(t)(y() y()) y(t)(x() x()). - 1 2. ,

F() =x()y() x()y() y()x() +y()x() =x()y() y()x()F() =x()y() x()y() y()x() +y()x() =x()y() +y()x()

..F() =F()

, (, ) :F

() = 0.

F

() =x

()(y() y()) y()(x() x()), .. x()(y() y()) =y ()(x() x()).2. x()= 0, x()=x(), .. .

1 .

[:|||||:]

, , . y(t) =t, . :

x() x()y() y()

=x()

y

()x() x()

=x()

x()

x() =x()(

)

x() =x(), .

x() x() =x()( ) = 0x() = 0

XIX

, f(x) E,

>0()> 0 :x, x E,|x x|< |f(x) f(x)|<

47

48/50

x0:

>0(, x0)> 0 :x E,|x x0|< |f(x) f(x0)|< , -

, . , , , x. -

x0, x .

45.14. f(x) E, f(x) E.

. x =x0, x =x, -

x0, x0 E. [:|||||:] 45.15. f(x) E, f(x) E E.

. . [:|||||:]

45.16. f(x) E, , , - E.

. -. f(x) = x2 E = (1, +). - :

0>0 : >0x, x (1, +), |x x|< |f(x) f(x)| 0 , x1 =x

, x2 = x.|f(x) f(x)|=|x21 x22|=|x1 x2||x1+ x2|.

, |x1 x2|= 2 . , |x1+ x2| . ,x1, x2(1, +) :|x1+x2|> 3 . , |x1 x2||x1+x2|> 32 =0. [:|||||:]

f(x2)

f(x2)

f(x2) +

f(x1)

f(x1) +

f(x1)

(x1 , x1 + ) (x2 , x2 + )

. f(x) = x2. , - , x - f(x) . f(x) , , - x - . , .

, , - .

45.17 ( ). f(x)C[a, b], f(x) [a, b].

. . 0 > 0 : >0x, x [a, b], |x x|< |f(x) f(x)| 0. = 1n , nN. nNxn, xn [a, b] :|xn xn| < 1n |f(xn) f(xn)| 0. .. xn, xn [a, b], {xn} {xn} , {xkn} {xn} {xkn} {xn}, lim

nxkn = , limn

xkn = . ..n N |xkn

xkn|< 1

n, =. xkn x

kn

[a, b], f(x) . , limn

f(xkn) =

f(), limn

f(xkn) =f() =f(). .. lim

nf(xkn) = limn

f(xkn). ,

|f(xn) f(xn)| 0. [:|||||:]

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XX

46

f(x), -E. ..f(x). f(x) . E, (f(x)) = f(x). , f(x) . E, f(x) = f(3)(x). n- f(n)(x) = (f(n1)(x)), f(0)(x) = f(x). -, n- .

, n E C(n)(E).

:

f(x) = sin x.

f(x) = cos x= sin

2+x

f(x) = sin x= sin (+x)f(x) = cos x= sin

3

2 +x

f(4)(x) = sin x

, f(n)(x) = sin

(n2

+x

). n = 1 .

n, n= n+ 1.

f(n+1)(x) = sin

(n+ 1)

2 +x

f(n+1)(x) = (f(n)(x)) =

sinn

2 +x

= cos

n2

+x

= sin

(n+ 1)

2 +x

f(x) =ex. f(n)(x) =ex.

f(x) =xm. n , f(n)(x) =m(m 1) . . . (m n + 1)xmn.

f(x) = ln x. f

(x) = 1x =x

1

. f(n)

(x1

) = (1)(2) . . . (n)x1n

= (1)n

n! x1n

. , f(n)(x) =f(n1)(x1) = (1)n1(n 1)!xn.

47

47.1. u(x) v(x) n E.

(u v)(n) =n

k=0

Ckn u(nk) v(k).

. . n = 1, (u

v) = uv+ uv =1

k=0 Ck1 u(nk) v(k). n, n = n+ 1.

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(u v)(n+1):

(u v)(n+1) =

nk=0

Ckn u(nk) v(k)

=n

k=0

Ckn u(nk+1) v(k) +n

k=0

kn u(nk) v(k+1) =

=

nk=0

Ckn u

(nk+1)

v(k)

+

n+1k=1

Ck1n u

(nk+1)

v(k)

=C0n u

(n+1)

v +n

k=1

(Ckn+Ck1n ) u(nk+1) v(k)++ Cnn u v(n+1) =C0n u(n+1) v +

nk=1

Ckn+1 u(nk+1) v(k) + Cnnu v(n+1) =n+1k=0

Ckn+1 u(nk+1) v(k)

:

Ckn+Ck1n =

n!

k!(n k)!+ n!

(k 1)!(n k+ 1)! =(n k+ 1)n! +kn!

k!(n k+ 1)! = (n+ 1)!

k!(n k+ 1)! =Ckn+1

[:|||||:]

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