Асимптотические оценки. Мицик М.Ф. - 1 стр.

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.
, , 
    , 
 .        
   .   ,  ,   
   , , ,  
.
,    ,  
    , 
    ,   , 
.
1
.  
( )
xf

( )
xg
   
( )
RdcS
⊂
=
;

( )
xg
  
) 
( )
xf

Sax
∈
, 
(
)
( )
1lim =
→
xg
xf
ax
.
 , 
( )
xg
 
( )
xf

a
x
→
, 
( ) ( )
xgxf ~
, 
( )
xf

( )
xg

a
x
→
.
 1. , 
1
sin
lim
0
=
→
x
x
x
,

xx ~sin

xsin

x

0
x
.
2
. 
()
xf
  
( )
xg

a
x
→
, 
( )
( )
xg
xf


a
x
=
; 
() ()( )
xgOxf =

a
x
.

(
)
( )
xg
xf

S
, 
() ()( )
xgOxf =

Sx
∈
.
 2.
( )( )
( )
=
+
+
−
=
−
→→
xx
xx
x
x
xx
cos1
cos1cos1
lim
cos1
lim
2
0
2
0
( )
( ) ( )
2
1
cos1
1
lim
cos1
sin
lim
cos1
cos1
lim
0
2
2
0
2
2
0
=
+
=
+
=
+
−
=
→→→
xxx
x
xx
x
xxx
,
                                                                                            .

                                                                            ,                   ,
                                                                                            ,
                           .
                                        .                               ,                           ,
                                                      ,                ,             ,
                                                          .
                   ,                                                                                    ,
                                                                                                                 ,
                                                                 ,                                           ,
                               .

          1.                                    f (x)         g(x)
S = (c; d ) ⊂ R                                  g (x)
                                                                                                 f (x)
                                             f ( x)       x → a∈S ,                       lim           = 1.
                                                                                                 g (x )
                       )                                                                  x→a


                       ,           g ( x)                                        f ( x)             x→a,
f ( x ) ~ g ( x)                    ,           f ( x)                                          g( x)                x→a.
                                                                                                                 sin x
                       1.                             ,                                                 lim            = 1,
                                                                                                            x →0   x
            sin x ~ x              sin x                                             x          x → 0.
          2.                       f (x )
                                                              f (x)
g (x)          x →a,
                                                              g (x )
                                                                 x = a;
                                            f ( x ) = O ( g ( x ))                       x→a.
                                              f (x )
                                              g (x )
S,                                 f ( x ) = O ( g (x ))                        x∈S .
                                   1 − cos x
                                             = lim
                                                   (1 − cos x)(1 + cos x) =
                       2. lim
                          x→0          x 2     x→0      x 2 (1 + cos x )
       1 − cos2 x             sin 2 x                1        1
= lim 2              = lim 2              = lim              = ,
  x→0 x (1 + cos x )   x→0 x (1 + cos x )   x→0 (1 + cos x )  2

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