3 Granične Vrednosti. Neprekidnost
October 1, 2022 | Author: Anonymous | Category: N/A
Short Description
Download 3 Granične Vrednosti. Neprekidnost...
Description
Ordgc ˏkge Ordgc ˏ kge vrejgfstc c gepreicjgfst Ordgc ˏkge Ordgc ˏ kge vrejgfstc lugikc`d u ^
tdˏkie kˏ ie d ∍ ^ fscb, bfŷjd, u Geid `e lugikc`d l (x) jefflgcsdgd u fifacgc U d td tdˏ tdkic kˏ ic d. Jefflgckc`d IfŤc`evd jefflgckc`d. Lugikc`d l (x) cbd
dif vdŷc
∈ ∋ ε 7 8
ordgckkgu ordgcˏ ˏ gu vrejgfst D ∍ ^ u ttddkkic ˏ ic d
| ∞ d| 1 κ ⇞ |l (x) ∞ D| 1 ε.
κ 7 8 8 1 x
Zdjd se pcŤe
acb l (x) 2 D
x↞d
Jruodˏkc`c Jruod kˏ c`c zdpcs0
∈ ∋ ε 7 8
Xrcber
>
acb x ↞4 x
∈ ∋ ε 7 8
∍ ∞ κ, d + κ ) Q {d} ⇞ l (x) ∍ (D ( D ∞ ε, D + ε).
κ 7 8 x (d ( d
κ 2 2
2 4, `er
∐ 4 + ε ∞ 4 7 7 8 8
| ∞ 4| 1 κ ⇞ |x> ∞ 4| 1 ε.
8 1 x
Pdcstd, geid `e ε 7 8 prfczvfa`gf. Cspctd`bf zd if`e vrejgfstc x vdŷc
| ∞ 4| 1 κ ⇞ |x> ∞ 4| 1 ε.
8 1 x
ZrdgslfrbcŤcbf czrdz
|x> ∞ 4| 2 |(x ∞ 4)( 4)(x x + 4)| 2 |(x ∞ 4)( 4)(x x ∞ 4 + >)| 2 |(x ∞ 4)> + >(x >(x ∞ 4)| ≪ |x ∞ 4|> + >|x ∞ 4| 1 ε . 4
> Uvfjeg`eb sbege t 2 |x ∞ 4| 7 7 8 8 pfsaejg`c usafv pfstd`e t> + >t >t ∞ ε 1 8 . ¯ >
>
t + >t ∞ ε 2 8 Idif `edif zdjfvfa`eg `e
⇚
t4,> 2
∐ : + :ε :ε
∞ ´ >
, trdŷeg trdŷegcc usafv `e
∐ ∐ | ∞ 4| 7 7 8 8 ∯ ∞4 ∞ 4 + ε 1 t 1 ∞4 + 4 + ε,
t 2 x
t`.
∞4 + ∐ 4 + ε ⇚
8 1 t 1
| ∞ 4| 1 ∞4 + ∐ 4 + ε
8 1 x
y 8.5 8.5
ϵ 2 2
8.; 8.;
2 ϵ 2
8.45 8.45
2 ϵ 2
4
F
? 8 . 8
: ; 4 . > . 8 8
κ
κ κ
4
2
2 2
Xrcber Dif `e acb l (x) 2 4, cspctdtc tdˏ tdkgfst kˏ gfst tvrjeg`d0 ¯ x↞8
• l (8) 7 (8) 7 > > 9 • ge pfstf`c vrejgfst l (8) (8)9 • ge pfstf`e vrejgfstc l (x) zd x ∍ (∞8.4, 8.4).
x
;
Xrcber Pd lugikc`u l (x) 2
gd gdºkkc º c acb l l ((x).
4
x, x 2 8 4, x 2 8
8
x↞8
Jefflgckc`d ordgcˏkgu ordgc kˏ gu vrejgfst D ∍ ^ u td tdˏkic kˏ ic d dif zd svdic gcz {xg }g∍G u U d Q {d} tdidv jd `e acb xg 2 d 2 d vdŷc Md`gefvd Md`gefv d jefflgckc`d. Lugikc`d l (x) cbd
g↞+∗
acb l (xg ) 2 D. g↞+∗
Xrcber
acb scg
x↞8
{x } ∍ {x } ∍
4 , acb xg 2 8, acb l ( l (xg ) 2 8, g↞+∗ g↞+∗ gψ > , acb xg 2 8, acb l (xg ) 2 4. G , xg 2 g↞+∗ g↞+∗ (:g (:g + 4)ψ 4)ψ
g g G,
g g
4 4 ge pfstf`c, `er zd l (x) 2 scg vdŷc x x
xg 2
y
F
x
: Xrcber
4 acb g scg 2 4 g↞+∗ g
^eŤeg`e0 Gd ordgckge kˏ ge vrejgfstc ^eŤeg`e0 Gd fsgfvu pfzgdte ordgcˏ scg x 24 x↞8 x acb
c Md`gefve jefflgckc`e, zd svdic gcz {xg }g∍G if`c zdjfvfa`dvd usafv acb xg 2 8 g↞+∗ vdŷc scg xg 2 4. g↞+∗ xg acb
Geid `e xg 2
4 g
↞ 8, g ↞ +∗. Zdjd `e acb
g↞+∗
scg g4 4 g
4 2 acb g scg 2 44.. g↞+∗ g
@ejgfstrdge ordgc ˏ ordgc ˏkge kge vrejgfstc
Jefflgckc`d Aevd ordgc ˏ ordgc ˏkgd kgd vrejgfst.
Geid `e lugikc`d l (x) jefflgcsdgd u cgtervdau (d ∞ κ 4 , d), d ∍ ^. Lugikc`d cbd aevu aevu ordgc ordgcˏkgu kˏ gu vrejgfst D4 ∍ ^ u td tdˏkic kˏ ic d dif vdŷc l (x) cbd
∈ ∋ ε 7 8
∞
⇞ |l (x) ∞ D4| 1 ε.
κ 7 8 8 1 d x 1 κ acb l (x) 2 D4 x↞d∞
Jefflgckc`d Jesgd ordgc ˏ ordgc ˏkgd kgd vrejgfst.
Geid `e lugikc`d l (x) jefflgcsdgd u cgtervdau (d, d + κ + κ 4 ), d ∍ ^. Lugikc`d tdˏkic kˏ ic d dif vdŷc jesgu ordgc ordgcˏkgu kˏ gu vrejgfst D> ∍ ^ u td cbd jesgu l (x) cbd
∈ ∋ ε 7 8
κ 7 8 8 1 x d 1 κ acb l (x) 2 D>
∞
x↞d+
⇞ |l (x) ∞ D>| 1 ε.
5 Zefrebd ˏ d ^ fscb, bfŷjd, bfŷjd, u tdˏ td kic ˏ d. Geid `e lugikc`d l (x) jefflgcsdgd u fifacgc U d td kie Ordgcˏ kgd vrejgfst pfstf`c dif c sdbf dif pfstf`e ordgcˏ kge vrejgfstc
∍
↞d l (x) xacb acb l (x) c acb l (x) x↞d+
x↞d∞
c `ejgdie su. Zdjd vdŷc
acb l (x) 2 acb l (x) 2 acb acb l (x).
x↞d
x↞d+
x↞d∞
Xrcber 4. l (x) 2 sog (x) 2 y
∞
4, x 7 8, 8 , 8, x 2 8, 4, x 1 8. 8 .
4
acb l (x) 2
∞4,
x↞8∞
acb l (x) 2 4,
x↞8+
x
F
acb l (x) ge pfstf`c.
- 4
.
>. l (x) 2 |x| 2
x↞8
≥
x, x 8, 8 , x, x 1 8. 8 .
∞
y
acb l (x) 2 ac acb b l (x) 2 8,
x↞8∞
x↞8+
acb l (x) 2 8. 8.
F
x
x↞8
< Ordgc ˏkge Ordgc ˏ kge vrejgfstc u ^
Jefflgckc`d acb l (x) 2 +
∗
x↞d
⇚ ∈B 7 8 ∋κ 7 8 acb l (x) 2 ∞∗ ∞∗ ↞ ⇚ ∈B 7 8 ∋κ 7 8
x
d
| ∞ d| 1 κ ⇞ l (x) 7 B.
8 1 x
8 1 x
| ∞ d| 1 κ ⇞ l (x) 1 ∞B.
Jefflgckc`d acb l (x) 2 D
x↞+∗
∈ε 7 8 ∋b 7 8
⇚
x7b
⇞ |l (x) ∞ D| 1 ε.
acb l (x) 2 D
x↞∞∗
⇚ Xrcber
∈ ∋ ε 7 8
b 7 8 x 1
4 x
4. l (x) 2 ,
∞b ⇞ |l (x) ∞ D| 1 ε.
x 2 8. 8.
y
acb l (x) 2
x↞8∞
F
⇞
x
∞∗,
acb l (x) 2 + , x↞8+
xacb ↞8 l (x) ge pfstf`c,
acb l (x) 2 8, acb l (x) 2 8.
x↞∞∗
x↞+∗
∗
? .
>. l (x) 2
4 , x>
x 2 8.
y
acb l (x) 2 ac acb b l (x) 2 +
x↞8∞
∗ ⇞ acb l (x) 2 +∗9 ↞8 x↞8+ x
acb l (x) 2 8, 8,
x↞∞∗
x
F
acb l (x) 2 8. 8.
x↞+∗
. 4
;. l (x) 2 e , x
x 2 8.
y
acb l (x) 2 8,
x↞8∞
acb l (x) 2 + ,
x↞8+
4
∗
acb l (x) ge pfstf`c9
x↞8
x
F
acb l (x) 2 4,
x↞∞∗
acb l (x) 2 4.
x↞+∗
Fsfncge ordgc ˏ ordgc ˏkgcm kgcm vrejgfstc
Zefrebd Dif `e δ
≪ l l ((x) ≪ ξ zd svdif x ∍ U Q {d} c dif pfstf`c acb l (x), tdjd `e ↞ δ ≪ acb l (x) ≪ ξ. ξ . ↞ d
x
d
x
d
= Zefrebd Dif `e l l ((x)
≪ o o((x) zd svdif x ∍ U Q{ Q {d} c dif pfstf`e acb l (x) c acb o (x), ↞ ↞ d
tdjd vdŷc
x
acb l (x)
d
x
d
≪ acb o (x). ↞
x↞d
x
d
Zefrebd Dif `e l (x)
≪ m( m (x) ≪ o o((x) zd svdif x ∍ U Q {d} c dif pfstf`e acb l (x) c ↞ d
x
d
acb o (x) c `ejgdie su, tdjd pfstf`c c acb m(x) c vdŷc
x↞d
x↞d
acb m(x) 2 acb l (x) 2 acb o (x).
x↞d
x↞d
x↞d
\dŷgf! Gdvejegd
tvrjeg`d (sd ≪) vdŷe c dif u usafvcbd vdŷe strfoe ¯ ge`ejgdifstc (1).
Zefrebd
Q {d} c acb o (x) 2 8, 8 , tdjd vdŷc ↞
[tdv Dif `e lugikc`d l (x) fordgcˏ kegd u U d
x
d
acb l (x) o (x) 2 8.
x↞d
Xrcber
acb x scg scg 4 4x 2 8, `er
x↞8
y
≪ scg
F
x
4 x
4, 4 ,
acb x 2 8.
x↞8
∈x 2 8,
3 Zefrebd Dif pfstf`e acb l (x) c acb o (x), tdjd pfstf`e c acb kl (x) x↞d
xacb ↞d
x↞d
x↞d
(k
∍ ^) ,
vdŷc ŷc l (x) + o (x) c xacb ↞d l (x) o (x) c vd
acb kl (x) 2 k k acb acb l (x),
x↞d
x↞d
acb l (x) + o (x) 2 acb l (x) + acb o (x),
x↞d
x↞d
x↞d
¿
acb l (x)o (x) 2 acb l (x) acb a cb o (x).
x↞d
x↞d
x↞d
l (x) c vd vdŷc ŷc x↞d o (x)
Dif `e `fŤ c acb o (x) 2 8, 8 , tdjd pfstf`c c acb
x↞d
acb l (x) l (x) x↞d acb 2 . x↞d o (x) acb o (x) x↞d
Zefrebd Dif pfstf`e pfs tf`e ordgcˏ kge vrejgfstc acb l (x) 2 n c acb L L ((y) x↞d
y ↞n
c dif zd svdif x U d d vdŷc l (x) 2 n, tdjd pfstf`c ordgcˏ kgd vrejgfst safŷege lugikc`e acb L l (x) c vdŷc
∍ Q { }
x↞d
acb L l (x) 2 acb L ( L (y ).
x↞d
y ↞n
Xrcber Uvfjeg`e sbege0 ¯
x>
• acb e ↞> x
• acb e ↞8+ x
4
x
2
↞ ⇞ ↞
2
x
t 2 x 2 x > t >
↞ x
:
2 acb et 2 e : . t↞:
⇞ ↞ ∗
4 t 2 x t 8+
+
2 acb et 2 + t↞+∗
∗.
48 Xrcberc ordgc ˏ ordgc ˏkgcm kgcm vrejgfstc scg x 24 x↞8 x acb
4+
acb
4
x↞+∗
x
2 e
x
scg x
4
x
scg; x x
Zefrebd scg x 2 4. x↞8 x acb
Jfidz0
• Jfidŷcbf gd`pre0
scg x 2 4. x↞8+ x acb
Cz pfzgdte ge`ejgdifstc y
scg x 1 x 1 tdg 1 tdg x, 8 1 x 1 ψ/> ψ/>,
saejc0
x
F
Pdtf
∞
scg x x
4 7 4 7 4 , tdg x scg x x scg x 4 7 7 kfs x, x scg x 8 1 4 1 4 1 4 kfs x. x
4 24
∞
scg x 1 4 x
vdŷc zd x> 1 > >ε, ε, t`. 8 1 x 1
∞
∞
∐ >ε.
> x
∞
x kfs x 2 > scg scg 1 > > >
>
x> 2 1ε >
44
∐ ∈ε 7 8 ∋κ 22 >ε 7 8 8 1 x 1 κ ⇞ ⇞ acb scg x 2 4. ↞8+ x
x
• Pnfo pdrgfstc lugikc`e l (x) 2 scgx x scg x • Idif `e acb 2 ↞8∞ x
1ε
scg x 2 4. x↞8∞ x
zdia`uˏku`e zdia`u kˏ u`e se jd `e c acb
scg x 2 4, tf `e x↞8+ x scg x 2 4. x↞8 x acb
∞4
acb acb
x
Xrcber
scg x x
scg dx d 2 , x↞8 scg nx n acb
n 2 8.
scg dx scg dx dx d ^eŤeg`e0 acb 2 acb dx 2 . x↞8 scg nx x↞8 scg nx n nx nx
¿ ¿
Xrcber
acb
4
x↞8
^eŤeg`e0 acb
∞ kfs ix 2 i> . x>
4
x↞8
>
∞ kfs ix x>
>scg> 2
acb
ix i x >
x>
x↞8
ix i x > ix
2
acb >
x↞8
¿
>
ix i x > > > i 2 i ix > :
scg
scg
¿
¿
>
Zefrebd acb
x↞+∗
• Geid `e { x } ∍
i i G
4+
4 x
x
2 e
prfczvfa`dg gcz tdidv jd `e acb xi 2 +∗ .
Jfidŷcbf jd `e acb
i↞+∗
i↞+∗
4+
4 xi
xi
2 e.
4>
• Fzgdˏ Fzgdkcbf0 kˏ cbf0 g 2 Yx R, i 2 4, >, . . . . Zdjd `e 4, g ≪ x 1 g + 4, i
i
i
i
i
fjdiae saeje ge`ejgdifstc0 4+
4 4 1 4 + gi + 4 xi
≪ 4 + g4 . i
≪ ≪ 4 4+ gi + 4
4+
4 xi
gi
4+
t`.
4 4 + gi + 4
• Idif `e g
gi
4+
4 xi
1 4+
4 xi
xi
gi +4
4 gi
G, g4 1 g> 1
i
i↞+∗
4 xi
gi +4
gi +4
,
,
4 1 4 + gi
xi
gi +4
.
4+
4 gi + 4
, c vdŷc
1 gi 1
gi
2
g 4 gi + 4 4 4+ gi + 4
i↞+∗
4+
4 gi
gi +4
i↞+∗
acb
2 acb
i ↞+∗
acb
i↞+∗
• Pdtf Ťtf `e {x } ∍
i i G
4+
4+
4 xi
i
4+
acb
i↞+∗
vdŷc c
,
∍ ¿ ¿¿¿ ¿ ¿ ¿ ¿
acb
acb
gi
1 4+
4+
4 1 4 + xi
gi
4 xi
4 gi
xi
gi
acb
i↞+∗
+4
2 e,
4+
4 gi
2 e,
2 e.
prfczvfa`dg gcz tdidv jd `e acb xi 2 +∗ , tf `e c acb
x↞+∗
4+
4 x
x
2 e.
i↞+∗
4; Gepreicjgfst lugikc`d
tdkie kˏ ie d ∍ ^. Geid `e lugikc`d l (x) jefflgcsdgd u fifacgc U d tdˏ Jefflgckc`d Lugikc`d l (x) `e gepreicjgd `e gepreicjgd u td tdˏkic kˏ ic d dif vdŷc
∈ ∋ | ∞ | ε 7 8
κ 7 8
x
d 1 κ
⇞ |l (x) ∞ l (d)| 1 ε.
Jruocb reˏ rekkcbd0 ˏ cbd0 Lugikc`d l (x) `e gepreicjgd u tdˏ tdkic kˏ ic d dif0 4. pfstf`c l (d), >. pfstf`c acb l (x), x↞d
;. acb l (x) 2 l l ((d). x↞d
y
Xrcber
4. Lugikc`d l (x) 2 x> `e gepreicjgd u tdˏ tdkic kˏ ic d 2 4, `er `e acb l (x) 2 4 2 l 2 l (4) (4)..
4
x↞4
F
y
4
>. Lugikc`d
4
l (x) 2 sog(x) 2 F
- 4
x
∞
4, x 7 8, 8 , 8, x 2 8, 4, x 1 8, 8 ,
gc`e gepreicjgd u d 2 d 2 8, `er acb l (x) ge pfstf`c. x↞8
x
4: y
;. Lugikc`d x, x
≥ 8, 8 ,
l (x) 2 x 2
| |
∞
x, x 1 8, 8 ,
`e gepreicjgd u d 2 d 2 8, `er `e F
x
acb l (x) 2 8 2 l 2 l (8) (8)..
x↞8
Jefflgckc`d Gepreicjgfst gd siupu Dif `e lugikc`d l (x) gepreicjgd u svdif` tdˏ tdkic kˏ ic d ∍ J , `e gepreicjgd gd siupu J . J ⊎ ^, fgd `e gepreicjgd K (J)
∞ siup lugikc`d gepreicjgcm gd siupu J siup lugikc`d gepreicjgcm gd seobegtu Yd, Y d, nR
K Yd, nR
Jefflgckc`d
∞
Lugikc`d l (x) `e gepreicjgd `e gepreicjgd zjesgd u zjesgd u tdˏ td kic kˏ ic d dif `e acb l (x) 2 l ( l (d). x↞d+ Lugikc`d l (x) `e gepreicjgd `e gepreicjgd saevd u saevd u tdˏ td kic kˏ ic n dif `e acb l (x) 2 l ( l (n). x↞n∞
td kic kˏ ic fnadstc jefflgcsdgfstc. \dŷgf! Lugikc`d `e gepreicjgd u czfafvdgf` tdˏ \dŷgf! [ve eaebegtdrge lugikc`e su gepreicjge u svf`cb fnadstcbd jefflgcsdgfstc.
Zefrebd
Dif su lugikc`e l (x) c o (x) gepreicjge gd seobegtu Yd, Y d, nR , tdjd `e c lugikc`d δl (x) + ξo( ξo (x) (δ, ξ ^) gepreicjgd gd Yd, Y d, nR.
∍ ∍
Zefrebd Dif su lugikc`e l (x) c o (x) gepreicjge gd seobegtu Yd, Y d, nR , tdjd `e c lugikc`d l (x)o (x) gepreicjgd gd Yd, Y d, nR. Dif `e o (x) 2 8 zd svdif x Yd, Y d, nR, tdjd `e c
l (x) lugikc`d gepreicjgd gd Yd, nR. o (x)
∍
45 Zefrebd Dif lugikc`d o (x) cbd ordgcˏ kgu vrejgfst acb o (x) 2 n c dif `e l (y ) x↞d
gepreicjgd u tdˏ kic y 2 2 n, n, tdjd safŷegd lugikc`d l o (x) cbd ordgcˏ kgu vrejgfst
acb l o (x) 2 l acb o (x) 2 l ( l (n).
x↞d
x↞d
Zefrebd Dif `e lugikc`d o (x) gepreicjgd u tdˏ kic x 2 d c lugikc`d l (y ) gepreicjgd u tdˏ kic y 2 n 2 o (d), tdjd `e c safŷegd lugikc`d l o (x) gepreicjgd u tdˏ kic
x 2 d. 2 d.
Xrcber
4. acb ag(4 + x ) 2 4. x
x↞8
4 t
Uvfjeg`eb sbege x 2 , t ↞ ∗ 0 ¯
ag(4 + x) 4 acb 2 acb acb t ag 4 + t↞∗ x↞8 x t
2 acb ag 4 + t↞∗
4 t
t
Idif `e ag t gepreicjgd lugikc`d, tf `e
acb ag 4 +
t↞∗
.
>. acb
x↞8
4 t
t
ex
2 ag ag
acb 4 +
t↞∗
4 t
t
2 ag e 2 4.
∞ 4 2 4.
x
[begfb ex ∞ 4 2 t, x 2 ag(4 + t), t ↞ 8 0 acb
x↞8
ex
∞ 4 2 acb
x
t↞8
t 2 ag(4 + t)
4 2 4. ag(4 + t) acb t↞8 t
.
4< Geie vdŷge ordgc ˏ ordgc ˏkge kge vrejgfstc0 vrejgfstc0
scg x 2 4, x↞8 x
acb
• •
acb
x↞+∗
4+
4 x
x
⇚
4
acb (4 + x) , x
x↞8
ag(4 + x) 2 4, x↞8 x
•
acb
•
acb
x↞8
ex
∞ 4 2 4.
x
Xrcber ag(4 + scg> x) ag(4 + scg> x) scg > x x> 2 4. 2 acb x↞8 x↞8 x> 4 ex 4 ex scg> x acb
>
¿
∞
¿
>
∞
\rste preicjd c gepreicjgf prfjuŷeg`e
Jefflgckc`d Geid `e lugikc`d l (x) jefflgcsdgd gd cgtervdau (δ, ξ ), fscb, bfŷjd u td tdˏkic kˏ ic
∍
d ( (δ, δ, ξ ). kˏ id preicjd lugikc`e preicjd lugikc`e l (x) dif vdŷc0 Zdkid kˏ id d (δ, `e tdˏkid ( δ, ξ ) `e td
∍
4. lugikc`d l (x) gc`e jefflgcsdgd u tdˏ tdkic kˏ ic d, cac >. lugikc`d l (x) `e jefflgcsdgd u tdˏ tdkic kˏ ic d , dac gc`e gepreicjgd u g`f`. 4 Xrcber Pd lugikc lugikc`u `u l (x) 2 tdˏ tdkid kˏ id x 2 8 `e tdˏ tdkid kˏ id preicjd, `er `e fgd x jefflgcsdgd gd ( ∞∗, 8) ∠ (8 (8,, +∗), t`. u svd svdif` if` ffifa ifacgc cgc td tdˏkie kˏ ie x 2 8 fscb u g`f` sdbf`.
∐
Pd lugikc` lugikc`uu l (x) 2 x tdˏ tdkid kˏ id x 2 8 gc`e tdˏ tdkid kˏ id preicjd, `er fgd gc`e jefflgcsdgd jefflgcsdgd gc u `ejgf` fifacgc td tdˏkie kˏ ie x 2 8. Zdifje, tdkid kˏ id x 2 ∞4 gc`e tdˏ tdkid kˏ id ¯ tdˏ preicjd, `er lugikc`d gc`e jefflgcsdgd gc u g`f` gc u g`egf` u fifacgc.
4? Geid `e x 2 tdkid kˏ id preicjd lugikc`e l (x). 2 d d tdˏ Jefflgckc`d Lugikc`d l (x) cbd preicj prve vrste u td tdˏkic kˏ ic d dif pfstf`e ifgdˏ ifgdkge kˏ ge ordgcˏ ordgckge kˏ ge vrejgfstc acb l (x) 2 n 4 c acb l (x) 2 n > . x↞d∞
x↞d+
Dif `e n4 2 n > , tdjd lugikc`d cbd ftiafg`cv preicj u td tdˏkkic ˏ ic d. Jefflgckc`d Lugikc`d l (x) cbd preicj jruoe vrste u tdˏ tdkic kˏ ic d dif d gc`e tdˏ tdkid kˏ id preicjd prve vrste, vrste, t`. dif `e ndr `ejgd fj ordgc ordgcˏkgcm kˏ gcm vrejgfstc acb l (x) c acb l (x) x↞d∞ x↞d+ nesifgdˏkgd nesifgd kˏ gd cac ge pfstf`c. Xrcber
4. l (x) 2 sog (x) 2 y 4
4, x 7 8, 8 , 8, x 2 8, 4, x 1 8, 8 ,
Zdkkid ˏ id x 2 8 `e tdˏ tdkkid ˏ id preicjd prve vrste, `er
∞4,
acb l (x) 2
x↞8∞
x
F
acb l (x) 2 4. x↞8+
Xreicj gc`e ftiafg`cv.
- 4
.
∞
>. l (x) 2 4 , x
y
x 2 8.
Zdkid kˏ id x 2 8 `e tdˏ tdkid kˏ id preicjd jruoe vrste, `er F
x
acb l (x) 2
x↞8∞
∞∗,
acb l (x) 2 + . x↞8+
∗
4= . 4
;. l (x) 2 e ,
x 2 8.
x
y
Zdkid kˏ id x 2 8 `e tdˏ tdkid kˏ id preicjd jruoe vrste, `er 4
acb l (x) 2 8, acb l (x) 2 +
x↞8∞
x↞8+
∗.
x
F
. :. l (x) 2 y
x> x
∞ 4 , ∞4
x 2 4.
Zdkid kˏ id x 2 4 `e tdˏ tdkid kˏ id preicjd prve vrste, `er
>
acb l (x) 2 acb l (x) 2 >.
4
x↞4+
x↞4∞ F
4
x
.
Xreicj `e ftiafg`cv.
4 x
5. l (x) 2 scg ,
x 2 8. 8.
y
Zdkkid ˏ id x 2 8 `e tdˏ tdkkid ˏ id preicjd jruoe vrste, `er F
acb l (x) c acb l (x)
x
x↞8∞
x↞8+
ge pfstf`e.
.
4 x
, x> `e gepreicjgf prfjuŷeg`e prfj uŷeg`e lugikc`e l (x) 2 x
4. Lugikc`d l ˑ(x) 2
>. Lugikc`d l ˑ(x) 2
∞ ∞
x 2 4, x 2 4, 4 gd ( 4
∞∗, +∗).
4 x scg , x 2 8, x 8, x 2 8,
4 x
`e gepreicjgf prfjuŷeg`e p rfjuŷeg`e lugikc`e l (x) 2 x scg gd (∞∗, +∗). @fŤ f gepreicjgfstc, td ˏ td ˏkidbd kidbd preicjd c ordgc ˏ ordgc ˏkgf` kgf` vrejgfstc
Xrcber Lugikc`d l (x) 2 tdg x,
y
x2
ψ >
+ iψ, i
P,
∍
`e gepreicjgd u keaf` fnadstc jefflgcsdgfs jefflgcsdgfstctc (idf eaebeg eaebegtdr tdrgd gd lug lugikc ikc`d) `d),, dac cbd nes nesifg ifgddkgf kˏ gf bgfof tdˏ tdkdid kˏ did preicjd jruoe vrste u td kidbd kˏ idbd
F
ψ + iψ, i >
∍ P.
Xrcber Cspctdtc gepreicjgfst lugikc`e l (x) 2 drktdg
4 x >x x>
| ∞∞
|
x
>8 c fjrejctc tdˏ tdkie kˏ ie c vrstu preicjd. ^eŤeg`e0 Lugikc`d `e jefflgcsdgd gd siupu Jl 2 (∞∗, 8) ∠ (>, ^eŤeg`e0 Lugikc`d (>, +∗) c bfŷe jd se prejstdvc u fnaciu l (x) 2
4∞x ∐ , >x ∞ x> 4∞x , drktdg ∐ x> ∞ >x drktdg
8 1 x 1 >, > ,
x 1 8
∭ x 7 >, > ,
lugikc`dd `e gepreicjgd idf ifbpfzckc`d eaebe eaebegtdrgcm gtdrgcm U keaf` fnadstc jefflgcsdgfstc Jl lugikc` lugikc`d. Zdkkie ˏ ie x 2 8 c x 2 > su tdˏ tdkkie ˏ ie preicjd. Idif vdŷc
| ∞∞ | ∗ ⇞ | ∞∞ | ∞∗ ⇞
acb
x↞8
↞> xacb
4 x 2+ x(> x)
4 x x(> x) 2
| ∞∞ | | ∞∞ |
acb drktdg x↞8
ψ 4 x 2 , > x(> x)
4 x xacb ↞> drktdg x(> x) 2
u g`cbd lugikc`d cbd ftiafg`cv preicj prve vrste.
Xrcber 4. Jcrcmae Jcrcmaefvd fvd lugikc` lugikc`dd Ϗ(x) 2
4, x T, 8, x ^ T,
∍ ∍ Q
`e jefflgcsdgd gd ^, dac gc`e gepreicjgd gc u `ejgf` td kic kˏ ic d ∍ ^.
∞
ψ >,
>4 >. Lugikc`d l (x) 2
∍ ∍ Q
x, x T, 8, x ^ T,
`e jefflgcsdgd gd ^, c gepreicjgd `e u 8. ;. Lugikc`d o(x) 2 x, x ∍ G, `e gepreicjgd u svcb tdˏ tdkidbd kˏ idbd fnadstc jefflgcsdgfstc `er su sve tdˏ tdkie kˏ ie x ∍ G czfafvdge. Jefflgckc`d (FpŤtc`d jefflgckc`d.) Geid `e lugikc`d l (x) jefflgcsdgd gd siupu J c geid `e d td t dkid kˏ id gdofbcadvdg`d siupd J . Zdjd0 acb l (x) 2 D
x↞d
∈ε 7 8
⇚ ∋κ 7 8 (∈x ∍ J J)) 8 1 |x ∞ d| 1 κ ⇞ |l (x) ∞ D| 1 ε.
:. Lugikc`d o(x) 2 4, x ∍ T `e gepreicjgd zd svdif s vdif x ∍ T. Fsfncge gepreicjgcm lugikc`d
Zefrebd Xrvd \d`erŤtrdsfvd tefrebd. Dif `e lugikc`d l (x) gepreicjgd gd seobegtu Yd, nR, tdjd `e fgd c fordgcˏ kegd0
∋ ∈ ∍ | { ∍ B 7 8
| ≪ B B..
x Yd, Y d, nR l (x)
fordgckkeg, ˏ eg, pd cbd svf` [iup vrejgfstc l Yd, nR 2 l (x) x Yd, nR} ⊎ ^ `e fordgcˏ cgfflbub c svf` suprebub. ŨtdvcŤe, pfstf`e tdˏ td kie kˏ ie u Yd, nR u if`cbd se fgc jfstcŷu. Zefrebd Jruod \d`erŤtrdsfvd tefrebd. Dif `e lugikc`d l (x) gepreicjgd gd seobegtu Yd, nR, tdjd pfstf`e tdˏ kie η, ο Yd, nR tdive jd vdŷc
∍
l l ((η ) 2 b b 2 cgl l (x), d≪x≪n
l (ο) 2 B B 2 sup l (x). d≪x≪n
>> Jfidz0 Lugikc`d l (x) `e gepreicjgd gd seobegtu Yd, nR, pd `e, prebd pretmfjgf` tefrebc, c fordgcˏ fordgckegd. kˏ egd. Zf zgdˏ zgdkc kˏ c jd pfstf`e B, b ∍ ^ tdivc jd `e B 2 sup l (x), x∍Yd,nR
b 2 cgl l (x). x∍Yd,nR
Jfidŷcbf jd pfstf`e tdˏ tdkie kˏ ie cz Yd, Yd, nR u if`cbd se jfstcŷu b c B . Xretpfstdvcbf jd `e l (x) 2 b zd svdif x ∍ Yd, Y d, nR. Zdjd, zd svdif x ∍ Yd, Y d, nR vdŷc0 l (x) 7 b
⇞ ⇞ ⇞
l (x) b 7 8 4 `e gepreicjgd gd seobegtu Yd, Yd, nR l (x) b 4 `e fordgc fordgcˏkegd. kˏ egd. l (x) b
∞
∞
∞
[tfod, pfstf`c ι tdif jd vdŷc 8 1
4 l (x) b
∞ ≪ ι ⇞
4 + b ι
≪ l ( l (x),
Ťtf `e suprftgf pretpfstdvkc jd `e b cgfflbub lugikc`e l . Xre Xrebd bd tfbe, tfbe, pfst pfstf`c f`c η ∍ ∍ Y Yd,d, nR tdif jd `e l l ((η ) 2 b. [ac [acˏkkgf ˏ gf se pfidzu`e c jd pfstf`c ο ∍ Yd, Y d, nR tdif jd `e l ( l (ο) 2 B. B . Gdpfbegd. Xretpfstdvid
f gepreicjgfstc lugikc`e gd seobegtu gd seobegtu `e `e gefpmfjgd.
Xrcber Lugikc`d l (x) 2 4 , x 2 >, `e gepreicjgd gd cgtervdau (8, (8, >), >), >∞x dac gd g`ebu gc`e fordgcˏ fordgckegd. kˏ egd. y
Zdifje, ¯ cdif `e
cgl l (x) 2 b b 2 2 8.5
B
81x1>
ifgdkdg ifgdˏ kˏ dg nrf`, cpdi ge pfstf`c x ∍ (8, (8, >) tdif jd `e l (x) 2 8. 8 .5.
b F
4.? >
x
Lugikc`d l (x) `e gepreicjgd gd svdifb seob seobeg egtu tu if if`c `c ge sd sdjr jrŷc ŷc >, gp gpr. gd tfb b se seob obeegtu Y8, Y8, 4.?R . Fgd `e gd tf fordgcˏkkegd fordgc ˏ egd c vdŷc 8.5 2 l (8) (8) x Y8, Y8 , 4.?R. ?R.
∈ ∍
≪ l (x) ≪ B B 2 l (4. (4.?), ?),
>; Zefrebd Xrvd Nfakdgf‛IfŤc`evd tefrebd. Dif `e lugikc`d l (x) gepreicjgd gd seobegtu Yd, nR c dif su l (d) c l (n) rdzacˏ kctfo zgdid, tdjd pfstf`c tdˏ kid η (d, ( d, n) tdivd jd `e l (η ) 2 8.
∍ ∍
Zefrebd Jruod Nfakdgf‛IfŤc`evd tefrebd. Dif `e lugikc`d l (x) gepreicjgd gd seobegtu Yd, nR c dif `e l (d) 1 l (n) , tdjd zd svdif K tdivf jd `e l (d) 1 K 1 l (n) pfstf`c tdˏ kid η Yd, Y d, nR tdive jd `e
∍ ∍
l (η ) 2 K.
Jfidz0 Pd lugikc` ikc`uu L ( vdŷe usaf usafvc vc zd prcbeg prcbeguu Xrv Xrvee Nfa Nfakdg kdgf‛I f‛IfŤc fŤc`e `eve ve Jfidz0 Pd lug L (x) 2 l l ((x) ∞ K vdŷe tefrebe, pd pfstf`c tdˏ tdkid kˏ id η ∍ ∍ (d, ( d, n) tdivd jd `e L ( L (η ) 2 8 ⇞ l (η ) 2 K. Gdpfbegd 4. Zvrjeg`e ¯ vdŷc c dif `e l (d) 7 l (n). Gdpfbegd >. Xretpfstdvid
f gepreicjgfstc lugikc`e gd seobegtu gd seobegtu `e `e gefpmfjgd.
Xrcber Lugikc`d l (x) 2
∞
∞ ∍ ∞
4, x 2 >, 4, x ( >, >R
`e jefflgcsdgd gd Y ∞>, >R, >R, vdŷc l l ((∞>) 2 ∞ 4 1 l (>) ( >) 2 4, dac gc`e gepreicjgd. Fgd jfstcŷe svf`u gd`bdg`u verjgfst ∞4 tdˏ tdkkic ˏ ic x gd`ve kku º u vrejgfst 4 td tdˏkkic ˏ ic x 2 2 ∞> c c svf`u gd`veº x 2 >. Cpdi, gc zd `ejgf K ∍ ∍ (∞4, 4) ge pfstf`c x ∍ Y∞>, >R tdif jd `e l ( l (x) 2 K. y 4
- > - 4
F
>
x
Xrcber Afidaczdkc`d guad lugikc`e l (x) 2 8.>x; + 8. 8.85 85x x>
∞ >.43; 43;x x + 4. 4.
>:
- :
4 F
- ;
> ;
x
\dŷgf! Pd lugikc`u l (x) if`d `e gepreicjgd gd seobegtu Yd, nR vdŷc0
ºu tdkie kˏ ie u Yd, nR u if`cbd lugikc`d jfstcŷe svf`u gd`bdg`u c svf`u gd`veº gd`vekku • pfstf`e tdˏ vrejgfst9
tdkid kˏ id u Yd, nR u if`f` lugikc`d jfstcŷe prfczv prfczvfa`gu fa`gu vrejgfst czbeju • pfstf`c tdˏ ¯ º e. gd`bdg`e c gd`veº gd`vekke.
[vf`u gd`bdg`u c svf`u gd`veº gd`vekku º u vrejgfst gd Yd, nR lugikc`d jfstcŷe u tdˏ tdkidbd kˏ idbd afidagcm eistrebubd cac u rungcb tdˏ tdkidbd kˏ idbd seobegtd.
y B
d
n x b
b 2 bcg l (x),
b 2 l 2 l ((n),
B 2 bdx l (x),
B 2 l ( l (η ),
d≪x≪n
d≪x≪n
∍ ∍
K Yb, Y b, B R,
l (k4 ) 2 l ( l (k> ) 2 K.
>5 Nrzcgd ifgveroegkc`e ifgveroegkc`e
Jefflgckc`d [cbnfac f c F .
Geid su lugikc`e l (x) c o(x) jefflgcsdge u geif` fifacgc tdˏ tdkie kˏ ie d ∍ ^, fscb, bfŷjd, u sdbf` tdˏ tdkkic ˏ ic d.
• l (x) 2 f(o(x)) idjd x ↞ d dif pfstf`c lugikc`d σ( σ (x) tdivd jd u geif` fifacgc tdˏ tdkie kˏ ie d vdŷc
l (x) 2 σ( σ (x)o (x)
(x 2 d) d ),
acb σ (x) 2 8.
x↞d
• l (x) 2 F(o(x)) idjd x ↞ d dif pfstf`c pfzctcvgd ifgstdgtd I tdivd tdivd jd u geif` fifacgc tdˏ tdkkie ˏ ie d vdŷc
|l (x)| ≪ I |o(x)|
↞ d ⇞
Gdpfbegd. l (x) 2 f o (x) , x
(x 2 d) d ).
↞ d.
l (x) 2 F o (x) , x
U prdisc ˏ prdisc ˏkkeŤº eŤkke º e ifrcstcbf saeje saejeºkke0 º e0 Jefflgckc`d Geid u tdˏ tdkkic ˏ ic d ∍ ^ pfstf`c ifgdˏ ifgdkkgd ˏ gd ordgc ordgcˏkkgd ˏ gd vrejgfst l (x) 2 A. x↞d o (x) acb
• Dif `e 8 8 1 1 |A| 1 1 + + ∗, idŷebf jd `e l l ((x) 2 F( F (o (x)) idjd x ↞ d . • Dif `e A A 2 2 8, tdjd idŷebf jd `e l l ((x) 2 f f((o (x)) idjd x ↞ d. • [pekc`dagf, dif `e A A 2 2 4, fgjd `e l l ((x) ∱ o o((x) idjd x ↞ d.
>< Xrcber scg x ↞8 x 2 4 4. xacb x5 2 8 x↞8 x;
>. acb
⇚
x
x5 2 f f((x; ),
x
>x; + 4 ;. acb 28 x↞+∗ x5
:. acb
x↞8
4
↞ 89
∱ x,
⇚
scg x
⇚
∞ kfs x 2 4 ⇚ x> >
↞ 89
>x; + 4 2 f f((x5 ), 4
∞ kfs x 2 2 F F((x> ),
↞ +∗9
x
↞ 89 .
x
Xrcber Pd lugikc`e l (x) 2 >x; scg 4 c o(x) 2 x ; ge pfstf`c acb l (x) 0 x↞8
x
o (x)
4 ; > x scg l (x) 4 x acb 2 acb 2 > acb scg . ; x↞8 o (x) x↞8 x↞8 x x
Cpdi, zd x 2 8 vdŷc
|>x; scg 4x | ≪ >|x|; ⇞ |l (x)| ≪ >|o(x)|, pd `e l (x) 2 F F((o (x)) , idjd x ↞ 8 . Gdpfbegd. l (x) 2 F F((o (x)) gc`e `ejgdifst u prdvfb sbcsau
>x> 2 F F((x> ), 5x> 2 F F((x> ),
↞ 8 ↞ 8
x x
>x> 2 5x> .
Jefflgckc`d Idŷebf jd `e lugikc`d l nesifgdˏ nesifgdkgf kˏ gf bdad veacˏ veackcgd cac kˏ cgd cac cgfflgctezcbdad cgfflgctezcbdad idjd idjd x ↞ d dif vdŷc acb l (x) 2 8.
x↞d
>? Xrcber Lugikc`d l `e `e nesifgdˏ nesifgdkgf kˏ gf bdad veacˏ veackcgd kˏ cgd idjd x ↞ d acb l (x) 2 8
⇚
x↞d
⇚
Jefflgckc`d
l (x) 28 x↞d 4
l (x) 2 f(4) f (4),, x
acb
⇚
d.
↞
Xfrejeg`e lugikc`d. ¯
Geid su l c c o nesifgd nesifgdˏkgf kˏ gf bdae veacˏ veackcge kˏ cge idjd x ↞ d .
• Dif vdŷc l l ((x) 2 f(o (x)) idjd x ↞ d, tdjd idŷebf jd `e l nesifgd nesifgdˏkgf kˏ gf rejd u u fjgfsu gd o . bdad veacˏ veackkcgd vcŤeo ˏ cgd vcŤeo rejd
• Dif vdŷc l l ((x) 2 F(o (x)) idjd x ↞ d, tdjd idŷebf jd su l c o nesifgdkgf kˏ gf bdae veacˏ veackcge cstfo kˏ cge cstfo rejd. rejd.
nesifgdkgf kˏ gf • Dif vdŷc l l ((x) 2 F F((o (x)) idjd x ↞ d , tdjd idŷebf jd `e l nesifgdˏ i
bdad veacˏ veackcgd rejd kˏ cgd rejd i u fjgfsu gd o .
• Dif `e l eivcvdaegtge. l ((x) ∱ o o((x) idjd x ↞ d, idŷebf jd su l c c o eivcvdaegtge. Xrditckgf kˏ gf `e pfrejctc "veacˏ "veackcgu" kˏ cgu" Gdpfbegd. Xrditcˏ
lugikc`e l ( l (x) sd geifb `ejgfstdvgfb lugikc`fb, gpr. o o((x) 2 (x ∞ d) , x ↞ d. i
Zefrebd g
g
• K ¿¿ f(x ) 2 f f((x ) , K ∍ ∍ ^ , • f(x )+ f(x ) 2 f(xbcg{ }) , • f(x ) ´ f(x ) 2 f f((x ) , • f(x ) ¿ f(x ) 2 f f((x + ) , • x ¿ f(x ) 2 f f((x + ) b
g
g
g
b
b
g
g
b,g
g
b g
b g
g
g
• K ¿¿ F(x ) 2 F( F (x ) , K ∍ ∍ ^ , • F(x )+F(x ) 2 F(xbcg{ }) , • F(x ) ´ F(x ) 2 F( F (x ) , • F(x ) ¿ F(x ) 2 F( F (x + ) , • x ¿ F(x ) 2 F( F (x + ) b
g
g
g
b
b
Xrcber Pd x ↞ 8 vdŷc0
g
g
x + >x >x> + 5x 5x; + F(x; ) + x: + F(x5 ) 2 x + >x >x> + F(x; ),
cac x + >x >x> + 5x 5x; + F(x; ) + x: + F(x5 ) 2 F F((x).
b,g
g
b g
b g
>= Xrcber Dif `e acb l (x) 2 >, cspctdtc tdˏ tdkgfst kˏ gfst gdvejegcm readkc`d idjd x ↞ 80 > x↞8
x
d) l l ((x) 2 f f((x> ),
n) l l ((x) 2 F F((x> ),
k) l ( l (x)
j) l l ((x) 2 f f((x; )
e) l l ((x) 2 f f((x),
l) x + l (x) 2 F( F (x)
m) l l ((x) 2 f f(4) (4),
c) xl xl ((x) 2 F( F (x; ).
∱ > >xx>
o) l l ((x)
^ezuatdt0 Zd kˏ gd su tvrjeg`d ^ezuatdt0 Zdˏkgd n), e), l), o), m), c). ¯
∱ x>,
View more...
Comments
