f:N->A ( ), A⊂R .
an
{an} -
.
A :
a1=1, a2=1, , an=1
an= , a1=1, a2= , , an= .
.
a {an}, n <
a=( ) = n
an= ; n=0 (a=0)
a n =| , n>
N=[ ]+1
, , .
- .
, a
(a<>liman)( =>|an-a|
an=(-1)n:-1;1;-1;1;
a (a- ,a+ )
Ʊ(a, ) .
a Ʊ(a, ) , , .
1)!!!
-:
()
, ; ,
,
Ʊ( ⋂ Ʊ( <> ∅
, ..
2) .
-:
A{ a N+1, aN+2, } ⊂ (a -1, a+1)
B={ a 1, a 2, , a N}
A∪B .
3) b>a => =>
c<a => =>
4) =>
5)
6)
()
.
{an} ,
{bn} (bn<>0) , { } .
:
{xn}, {yn} , , , {xn+yn}, { xn-yn }, { xnyn }, { xn/yn } (yn<>0)
!!! { n} { n} , , .
!!!
-:
{xn}
!!!
1) ,
2) ,
3)
-:2)
.
{an} (),
an<an+1 (an>an+1)
.
{an} ( ), an an+1 (an an+1)
!!! . , .
{an}
!!! ( )
!!!
|
|
!!!()
, .
.
f(x) ()
(x0)={x:D<|x-x0|<δ()}
1. () a f(x) x0 ( {xn}) ,
{ =x0, xn<>x0} => =a
2 (). a f(x) x0, ( >0) ()
( x:0<|x-x0|< óx (x0, )) => (|f(x)-a|< ó f(x) (a,
!!!
f(x) (x0-c,x0)
. a f(x) x0 x -> x0 ( ) ,
) ( {xn}), xn->x0
(xn<x0)
) ()()
( x:0<x0-x< => |f(x)-a|<
. (. )
!!! .
!!! f(x) , .
a= ( )
a= =f(x0-0) ( )
a= =f(x0+0) ( )
,
1. ,
2. , .
3. , b<a (b>a), ( (x0))( 0)) => f(x)>b (f(x)<b))
4. =a ( 0))(f(x) => a
5. 0))
( = =a)
6.
() 0) =y0 , x Ʊ(x0) f(x) <>y0
F(y) (y0) =>
!!!
1.
2.
3. , b<>0
x0 (x->x0)
(. .) (x->x0)
!!! a= óf(x)-a (x->x0)
f(x) 0
) ( xn)(xn -> x0; xn<>x0) => f(x0) ->
) ( ( (x0, => |f(x)|>M, f(x)>M,
f(x)<(-M)
f(x) (x->x0)
. x0
. x0=
!!! ó (
( x0, |f(x)-f(x)|<
f(x) a,b
1) x1<x2 => f(x1)<f(x2)
2) x1 x2 => f(x1) f(x2)
3) x1>x2 => f(x1)>f(x2)
4) x1 x2 => f(x1) f(x2)
!!! f(x) (a,b) , . . |f(x)| k, (
0, 0+0) 0-0)
, .
f(x) g(x) 0)
0)
1)|f(x)|<c|g(x)| (c>0)
f(x)=0(g(x)), x 0)
2)f, g x->x0
f(x)=0(g(x)), x->x0
x3=0(x2), x->0