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2
cos1
x
x
−
0
=
x
,
( )
2
cos1 xOx =−
,
0
→
x
.
,
ε≥x
222
2
cos1
cos1
ε
≥
+
≥
−
x
x
x
x
,
{ }
ε<∈ xRx \
2
cos1
x
x
−
,
( )
2
cos1 xOx =−
{ }
0\Rx
∈
.
3
.
()
xf
( )
xg
a
x
→
,
( )
( )
0lim =
→
xg
xf
ax
.
() ()( )
xgoxf
=
,
a
x
→
.
3.
;
ln
1
lim
.
ln
1
,0,0
lim
0
1
1
0
2
2
t
t
t
x
xte
x
e
t
x
x
x
−
=
−
=
→→=
=
→
−
−
→
0lim
2
1
0
=
−
→
x
e
x
x
( )
xoe
x
=
−
2
1
,
0
→
x
.
.
a
1, 2, 3 ,
,
x
, ,
.
.
4
. ,
( ) ( ) ( ) ( )
,....,...,,,
210
xxxx
n
ψ
ψ
ψ
ψ
,
....,2,1,0
=
n
( ) ( )( )
xx
nn
ψ
ψ=
+1
,
∞
→
x
.
( )
,0
2
lim
2
1
lim
1
ln
lim
1
ln
lim
ln
1
lim
2
0
3
0
2
0
2
0
2
0
==
−
−
=
′
′
−
=
−
=
−
→→→→→
t
t
t
t
t
t
t
t
t
ttttt
1 − cos x
x = 0,
x2
1 − cos x = O x 2 , ( ) x → 0.
1 − cos x 1 + cos x 2
, x ≥ε ≥ ≥ 2,
x 2
x 2
ε
x∈ R \ { x < ε} 1 − cos x
x2
, 1 − cos x = O x 2 ( ) x ∈ R \ {0} .
3. f (x )
f (x)
g ( x) x →a, lim = 0.
x→a g (x )
f ( x ) = o ( g ( x )) , x→a.
1
1 −
−
e x2 e x2
= t → 0, x → 0, t
lim = = lim ;
3. x→0 x 1 t→ 0 1
x= .
− ln t − ln t
1
′ −
t 2
− ln t − (ln t ) t t2
lim = lim = lim = lim = lim = 0,
t→0 1 t→0 1 t→0 ′ t→0 − 2 t →0 2
1
− ln t t2 2 t3
t
1
−
x2 1
e −
lim
x→ 0
=0 e x2
= o( x ) , x → 0.
x
. a 1, 2, 3 ,
, x
, ,
.
.
4. ,
ψ 0 ( x ),ψ 1 ( x ),ψ 2 (x ),...,ψ n (x ),.... , n = 0,1, 2, ....
ψ n+1 ( x ) = (ψ n ( x )) , x → ∞.
