Методическое пособие по теме "Нахождение пределов"

ÌÅÒÎÄÈ×ÅÑÊÈÅ ÓÊÀÇÀÍÈß
ïî òåìå
Íàõîæäåíèå ïðåäåëîâ
Ì î ñ ê â à
2 0 0 4
ÌÈÍÈÑÒÅÐÑÒÂÎ ÎÁÐÀÇÎÂÀÍÈß ÐÎÑÑÈÉÑÊÎÉ ÔÅÄÅÐÀÖÈÈ
ÌÈÍÈÑÒÅÐÑÒÂÎ ÐÎÑÑÈÉÑÊÎÉ ÔÅÄÅÐÀÖÈÈ
ÏÎ ÀÒÎÌÍÎÉ ÝÍÅÐÃÈÈ
ÌÎÑÊÎÂÑÊÈÉ ÈÍÆÅÍÅÐÍÎ-ÔÈÇÈ×ÅÑÊÈÉ ÈÍÑÒÈÒÓÒ
(ÃÎÑÓÄÀÐÑÒÂÅÍÍÛÉ ÓÍÈÂÅÐÑÈÒÅÒ)
ÌÅÒÎÄÈ×ÅÑÊÈÅ ÓÊÀÇÀÍÈß
ïî òåìå
Íàõîæäåíèå ïðåäåëîâ
Ìîñêâà
2004
ÓÄÊ 519.2(07)
ÁÁÊ 22.171ÿ7
Ì 54
Ìåòîäè÷åñêèå óêàçàíèÿ ïî òåìå Íàõîæäåíèå ïðåäåëîâ.
Ì.: ÌÈÔÈ, 2004. 25 ñ.
Ðàññìîòðåíû íåêîòîðûå ñïîñîáû ðåøåíèÿ çàäà÷, ïðåäëàãàåìûõ ñòóäåíòàì ïåðâîãî ñåìåñòðà
âñåõ ôàêóëüòåòîâ â äîìàøíåì çàäàíèè ÄÇ 27:
íàõîæäåíèå ïðåäåëîâ è âûäåëåíèå ãëàâíûõ ÷ëåíîâ ó ÷èñëîâûõ ïîñëåäîâàòåëüíîñòåé è ôóíêöèé
îäíîé ïåðåìåííîé. Ïðèâåäåíà êðàòêàÿ òàáëèöà
ñâîéñòâ ýêâèâàëåíòíûõ âåëè÷èí. Äàíî 30 ïðèìåðíî îäèíàêîâûõ ïî òðóäíîñòè âàðèàíòîâ äîìàøíèõ çàäàíèé.
Ïðåäíàçíà÷åíû äëÿ ñòóäåíòîâ ïåðâîãî êóðñà âñåõ ôàêóëüòåòîâ.
Àâòîðû: À.Ï. Ãîðÿ÷åâ, Þ.Í. Ãîðäååâ, Ä.Ñ. Òåëÿêîâñêèé/
Ïîä ðåäàêöèåé äîöåíòà À.Ï. Ãîðÿ÷åâà.
Ðåêîìåíäîâàíî ê èçäàíèþ ðåäñîâåòîì ÌÈÔÈ
c Ìîñêîâñêèé èíæåíåðíî-ôèçè÷åñêèé èíñòèòóò
(ãîñóäàðñòâåííûé óíèâåðñèòåò), 2004 ã.
1. Âû÷èñëåíèå ïðåäåëà
ïîñëåäîâàòåëüíîñòè
1.1. Ïðèìåð ðåøåíèÿ çàäà÷è
Íàéòè ïðåäåë ïîñëåäîâàòåëüíîñòè èëè äîêàçàòü, ÷òî
îí íå ñóùåñòâóåò.
Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü {xn }∞
n=0 , çàäàííóþ ðåêóððåíòíûì ñîîòíîøåíèåì:
r
x0 = 0;
xn+1 =
x2n + 4xn + 5
, n = 0, 1, 2, . . . .
10
(1.1)
Äîêàæåì, ïîëüçóÿñü ìåòîäîì ìàòåìàòè÷åñêîé èíäóêöèè,
÷òî ïîñëåäîâàòåëüíîñòü (1.1) îãðàíè÷åíà, à èìåííî:
0 6 xn < 1.
(1.2)
Ëåâîå èç íåðàâåíñòâ (1.2) âûïîëíÿåòñÿ ñîãëàñíî îïðåäåëåíèþ ïîñëåäîâàòåëüíîñòè (1.1), îòêóäà òàêæå âûòåêàåò è
ïðàâîå èç íåðàâåíñòâ (1.2) äëÿ n = 0. Ïðåäïîëîæèì, ÷òî
íåðàâåíñòâî (1.2) ñïðàâåäëèâî äëÿ íåêîòîðîãî íàòóðàëüíîãî n, è óñòàíîâèì, ÷òî îíî áóäåò âûïîëíÿòüñÿ è äëÿ n + 1.
Äåéñòâèòåëüíî,
x2n+1 − 1
(xn − 1) (xn + 5)
=
< 0,
xn+1 − 1 =
xn+1 + 1
10 (xn+1 + 1)
òî åñòü
xn+1 < 1.
Òåì ñàìûì íåðàâåíñòâî (1.2) ïîëíîñòüþ äîêàçàíî äëÿ
âñåõ n = 1, 2, 3, . . . .
3
Ïîêàæåì, ÷òî ïîñëåäîâàòåëüíîñòü (1.1) ÿâëÿåòñÿ ìîíîòîííîé. Ðàçíîñòü
x2n+1 − x2n
(1 − xn ) (9xn + 5)
=
,
xn+1 − xn =
xn+1 + xn
10 (xn+1 + xn )
è ïîñêîëüêó xn < 1 ñîãëàñíî (1.2), òî xn+1 > xn , òî åñòü
ïîñëåäîâàòåëüíîñòü {xn }∞
n=0 ÿâëÿåòñÿ ìîíîòîííî âîçðàñòàþùåé.
Òàêèì îáðàçîì, ïîñëåäîâàòåëüíîñòü (1.1) ìîíîòîííà è
îãðàíè÷åíà. Ñëåäîâàòåëüíî, ïî òåîðåìå î ñõîäèìîñòè ìîíîòîííûõ è îãðàíè÷åííûõ ïîñëåäîâàòåëüíîñòåé ñóùåñòâóåò
ïðåäåë
b = lim xn .
(1.3)
n→∞
Îòìåòèì, ÷òî èç (1.1) è (1.3) ñîãëàñíî òåîðåìàì î ïðåäåëüíîì ïåðåõîäå â íåðàâåíñòâàõ ñëåäóåò, ÷òî
(1.4)
0 6 b 6 1.
Äëÿ íàõîæäåíèÿ b ðåêóððåíòíóþ ôîðìóëó (1.1) çàïèøåì
â âèäå
10x2n+1 = x2n + 4xn + 5.
Ïåðåõîäÿ â ýòîì ðàâåíñòâå ê ïðåäåëó, ïîëó÷èì:
10 · lim xn+1 · lim xn+1 = lim xn · lim xn + 4 lim xn + 5.
n→∞
n→∞
n→∞
n→∞
n→∞
Òàê êàê b = lim xn = lim xn+1 , òî âåëè÷èíà b óäîâëån→∞
n→∞
òâîðÿåò êâàäðàòíîìó óðàâíåíèþ
5
10b2 = b2 +4b+5, ⇐⇒ 9b2 −4b−5 = 0, ⇐⇒ b = 1 èëè b = − .
9
Îòñþäà è èç (1.4) ñëåäóåò, ÷òî b = 1. Èòàê,
lim xn = 1.
n→∞
4
1.2. Âàðèàíòû çàäàíèé. Íàéòè ïðåäåë
ïîñëåäîâàòåëüíîñòè èëè äîêàçàòü,
÷òî îí íå ñóùåñòâóåò

1
x0 = 0,
2
x0 = 0,
3
x0 = 0,
4
x0 =
1
,
2
5
x0 =
1
,
3
Ïîñëåäîâàòåëüíîñòü
r
x2n + xn + 1
xn+1 =
,
3
r
x2n + 2xn + 3
xn+1 =
,
6
r
x2n + xn + 2
xn+1 =
,
4
r
3x2n + 2xn + 1
xn+1 =
,
6
r
3x2n + 1
xn+1 =
,
4
1
x0 = − , xn+1 = xn + x2n + x3n ,
2
1
7 x0 = ,
xn+1 = xn − x2n ,
2
r
x2n + xn + 1
5
xn+1 =
,
8 x0 = ,
2
3
r
x2n + 2xn + 3
,
9 x0 = 2,
xn+1 =
6
r
x2n + xn + 2
10 x0 = 2,
xn+1 =
,
4
6
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
ïðîäîëæåíèå íà ñëåäóþùåé ñòðàíèöå
5
ïðîäîëæåíèå

11 x0 = 3,
12 x0 = 2,
Ïîñëåäîâàòåëüíîñòü
r
3x2n + 2xn + 1
,
xn+1 =
6
r
3x2n + 1
xn+1 =
,
4
1
13 x0 = − , xn+1 = xn + 2x2n + 3x3n ,
3
p
1
xn+1 = xn − x2n ,
14 x0 = ,
4
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
15 x0 = 0,
xn+1 =
x2n + xn + 1
,
3
n = 0, 1, 2, . . .
16 x0 = 0,
xn+1 =
x2n + 2xn + 3
,
6
n = 0, 1, 2, . . .
x2n + xn + 2
=
,
4
17 x0 = 0,
xn+1
18 x0 = 0,
xn+1 =
3x2n + 2xn + 1
,
6
n = 0, 1, 2, . . .
19 x0 = 0,
xn+1 =
3x2n + 1
,
4
n = 0, 1, 2, . . .
1
20 x0 = − , xn+1 = xn + 3x2n + 2x3n ,
3
p
3
21 x0 = ,
xn+1 = xn − x2n ,
5
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
îêîí÷àíèå íà ñëåäóþùåé ñòðàíèöå
6
îêîí÷àíèå

Ïîñëåäîâàòåëüíîñòü
22 x0 = 2,
3
23 x0 = ,
2
xn+1 =
xn+1
x2n + 2xn + 3
,
6
x2n + xn + 2
=
,
4
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
24 x0 =
3
,
4
xn+1 =
3x2n + 2xn + 1
,
6
n = 0, 1, 2, . . .
25 x0 =
1
,
2
xn+1 =
3x2n + 1
,
4
n = 0, 1, 2, . . .
1
26 x0 = − , xn+1 = xn + x2n + 2x3n ,
3
n = 0, 1, 2, . . .
1
27 x0 = − , xn+1 = xn + 2x2n ,
3
n = 0, 1, 2, . . .
1
28 x0 = − , xn+1 = xn + 4x2n + 2x3n ,
4
n = 0, 1, 2, . . .
29 x0 =
1
,
2
xn+1 = xn + x2n − x3n ,
30 x0 =
1
,
2
xn+1 =
p
2 − x2n ,
n = 0, 1, 2, . . .
n = 0, 1, 2, . . .
2. Âû÷èñëåíèå ïðåäåëà ôóíêöèè
Äëÿ íàõîæäåíèÿ ïðåäåëîâ ôóíêöèé (à òàêæå äëÿ âûäåëåíèÿ ãëàâíûõ ÷ëåíîâ ïîñëåäîâàòåëüíîñòåé è ôóíêöèé)
íàì ïîòðåáóþòñÿ àñèìïòîòè÷åñêèå ðàçëîæåíèÿ íåêîòîðûõ
7
îñíîâíûõ ýëåìåíòàðíûõ ôóíêöèé ïðè x → 0 (òàáë. 1), à
òàêæå ñâîéñòâà ñèìâîëà o ìàëîå (òàáë. 2).
Òàáëèöà 1
(1 + x)α = 1 + αx + o(x)
sin x = x + o(x)
cos x = 1 −
x2
+ o x2
2
tg x = x + o(x)
ln(1 + x) = x + o(x)
sh x = x + o(x)
ex = 1 + x + o(x)
ch x = 1 +
ax = 1 + x ln a + o(x)
x2
+ o x2
2
th x = x + o(x)
Ïóñòü C íåêîòîðàÿ ïîñòîÿííàÿ, íå ðàâíàÿ íóëþ, à α, β
è γ ïðîèçâîëüíûå âåùåñòâåííûå ÷èñëà. Òîãäà ñïðàâåäëèâû
ñîîòíîøåíèÿ:
Òàáëèöà 2
o(f ) ± o(f ) = o(f )
o(Cf ) = o(f )
Co(f ) = o(f )
o(o(f )) = o(f )
γ
fα · o fβ
= o f (α+β)γ
o(f + o(f )) = o(f )
8
2.1. Ïðèìåð ðåøåíèÿ çàäà÷è
Íàéòè ïðåäåë ôóíêöèè èëè äîêàçàòü, ÷òî îí íå ñóùåñòâóåò :
lim (cos x)ctg
x→0
2
x
(2.1)
.
 ýòîì ïðèìåðå ìû èìååì äåëî ñ íåîïðåäåëåííîñòüþ âè2
2
äà 1∞ . Òàê êàê (cos x)ctg x = ectg x·ln cos x , òî âîñïîëüçîâàâøèñü íåïðåðûâíîñòüþ ïîêàçàòåëüíîé ôóíêöèè, áóäåì èñêàòü ïðåäåë ïîêàçàòåëÿ. Ñîãëàñíî ïðèâåä¼ííûì âûøå òàáëèöàì ïðè x → 0 èìååì:
x2
1
2
2
=
+o x
ctg x · ln cos x = 2 ln 1 −
tg x
2
x2
x2
1 o (x2 )
− + o x2
− + o x2
− +
2
x2 → − 1 ,
= 2
= 2
2 =
2
2
o (x2 )
x + o (x )
2
(x + o(x))
1+
2
x
1
1
ctg2 x
òî åñòü ïðåäåë (2.1) ðàâåí lim (cos x)
= e− 2 = √ .
x→0
e
2.2. Âàðèàíòû çàäàíèé. Íàéòè ïðåäåë
ôóíêöèè èëè äîêàçàòü, ÷òî îí íå
ñóùåñòâóåò
1. lim 23x + 32x − cos x
x→0
2. lim (2x − sin 2x)ctg x .
x→0
ctg x
1
2x
.
2 + 33x + 45x sin x
.
4. lim
x→0
3
ctg 2x
.
5. lim ex − e2x + e3x
x→0
3. lim
x→0
2x + 3x
2
2 ctg x
.
6. lim 23x + 3 sin 2x
x→0
9
ctg 3x
.
7. lim
ex + cos x + cos 2x
3
3x + cos 3x + sin 2x
2
x→0
8. lim
x→0
9. lim 2x −32x +cos 3x
3
sin x
2
x
.
.
1
x→0
x.
10. lim (2x + 3x − 4x )ctg x .
x→0
x
22
11. lim
x→0
12. lim
x→0
+
2
x
33
!
12
sin x
.
e2x + e3x + e4x
3
13. lim 22x + 2 sin x
ctg 2x
ctg 2x
x→0
14. lim
ex + cos 2x
2
3x + cos x − sin 2x
2
ex + e3x
2
2x + 22x + 23x
3
x→0
16. lim
x→0
17. lim
x→0
.
x→0
15. lim
2
x
cos 2x
sin x
.
.
2
sin x
.
ctg x
.
10
.
18. lim
x→0
2x + cos 2x + sin x
2
19. lim 23x +32x −4x
cos 3x
sin x
x→0
x
23
20. lim
x→0
21. lim
x→0
+
2
x
32
x→0
2x + 3x + cos x
3
25. lim
x→0
.
3
x
.
1
sin x .
22x + 33x + cos 4x
3
24. lim e2x −2 sin 3x
x→0
.
.
x→0
23. lim
x
!24 ctg 2x
22. lim 23x − 32x + 4x
2
3
sin x
.
1
sin 2x .
24x + 33x + cos 4x
3
cos 2x
sin x
cos 2x
sin 3x sin x
x
.
26. lim e +
x→0
3
27. lim (ex−cos x+cos 2x)ctg x.
x→0
1
sin 2x sin 2x
3x
28. lim e −
.
x→0
2
11
.
2
ctg x
30. lim ex − tg x
.
29. lim (ex + sin x)ctg 2x .
x→0
x→0
3. Âûäåëåíèå ãëàâíîãî ÷ëåíà
ôóíêöèè
3.1. Ïðèìåð ðåøåíèÿ çàäà÷è
Íàéòè äëÿ ôóíêöèè
äà
f (x)
ïðè
α
C (x − x0 ) :
f (x) = (cos 2x)3 tg
2
x
x → x0
− 1,
ãëàâíûé ÷ëåí âè-
åñëè x → 0.
(3.1)
Çàïèøåì ôóíêöèþ (3.1) â âèäå
f (x) = e3 tg
2
x·ln cos 2x
−1
(3.2)
è, âîñïîëüçîâàâøèñü òàáëèöàìè 1 è 2, ïðåîáðàçóåì ïîêàçàòåëü ó ýêñïîíåíòû â âûðàæåíèè (3.2). Ìû èìååì:
2
3 tg2 x · ln cos 2x = 3 x + o(x) ln 1 − 2x2 + o (x2 ) =
= 3 x2 + o (x2 ) −2x2 + o (x2 ) =
= 3 −2x4 + o (x4 ) = −6x4 + o (x4 ) .
Îòñþäà è èç (3.2) ïîëó÷àåì
f (x) = e−6x
4 +o
(x4 ) −1 =
= −6x4 + o (x4 ) + o −6x4 + o (x4 ) = −6x4 + o (x4 ) ,
òî åñòü
f (x) − g(x) = o x4 ,
12
ãäå g(x) = −6x4 . Ñëåäîâàòåëüíî,
f (x) ∼ −6x4 .
Òàêèì îáðàçîì, C = −6, à α = 4.
3.2. Âàðèàíòû çàäàíèé
Íàéòè äëÿ ôóíêöèè
÷ëåí âèäà
f (x)
α
ïðè
x → x0
ãëàâíûé
C (x − x0 )
1. f (x) = (cos x)cos x − 1,
x → 0.
2. f (x) = xctg x − 1,
x → 1.
sin x
3. f (x) = (cos x)
− 1,
x → 0.
π
x→ .
2
π
x→ .
2
π
x→ .
4
π
x→ .
2
x → 0.
π
x→ .
4
x → 1.
π
x→ .
2
x → 2π.
π
x→ .
4
4. f (x) = (sin x)x − 1,
5. f (x) = (sin x)sin x − 1,
6. f (x) = xtg x −
π
,
4
7. f (x) = (sin x)ctg x − 1,
8. f (x) = (cos x)x − 1,
9. f (x) = (tg x)tg x − 1,
10. f (x) = xsin x − 1,
11. f (x) = (sin x)cos x − 1,
12. f (x) = (cos x)x − 1,
13. f (x) = (ctg x)ctg x − 1,
13
1
,
π
15. f (x) = (tg x)ctg x − 1,
14. f (x) = xcos x −
x → π.
π
x→ .
4
π
x→ .
4
x → 1.
π
x→ .
4
π
x→ .
2
π
x→ .
2
π
x→ .
4
x → 1.
16. f (x) = (tg x)x − 1,
17. f (x) = xcos x − 1,
18. f (x) = (ctg x)tg x − 1,
19. f (x) = xsin x −
π
,
2
20. f (x) = xcos x − 1,
21. f (x) = (ctg x)x − 1,
22. f (x) = xtg x − 1,
23. f (x) = (cos x)tg x − 1,
2
,
24. f (x) = xsin x −
3π
26. f (x) = xsin x − 1,
x → 0.
3π
x→
.
2
3π
.
x→
2
x → π.
27. f (x) = xcos x − 2π,
x → 2π.
25. f (x) = xcos x − 1,
28. f (x) = x
sin x
− 1,
x → 2π.
π
x→ .
2
5π
.
x→
4
29. f (x) = xctg x − 1,
30. f (x) = (tg x)ctg x − 1,
14
Íàéòè äëÿ ôóíêöèè
÷ëåí âèäà
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
f (x)
ïðè
x → x0
ãëàâíûé
C
(x − x0 )α
√
x+1−1
f (x) =
,
ln cos x √
1 − cos x · cos 2x
f (x) =
,
sin x3
1 + sin2 x − ch x
,
f (x) =
ex − cos x
1
,
f (x) =
ln tg x
tg x
,
f (x) = √
3
1 − sin x
ln x
f (x) = 3
,
x − 3x + 2
√
ctg 3 x
√ ,
f (x) =
1 − cos 3 x
1
√
f (x) = √
,
1 − 2x − 3 1 − 3x
arcsin x
f (x) = x2
,
e − cos x
sh x
,
f (x) =
2 sin x − tg 2x
tg x
√
f (x) = √
,
ch x − 1 − x2
√
√
x−1− 3x−1
f (x) =
,
sin2 πx
1
√
,
f (x) =
sin x2 + 5 − 3
15
x → 0.
x → 0.
x → 0.
π
.
4
π
x→ .
2
x→
x → 1.
x → 0.
x → 0.
x → 0.
x → 0.
x → 0.
x → 2.
x → 2.
14. f (x) =
15. f (x) =
16. f (x) =
17. f (x) =
18. f (x) =
19. f (x) =
20. f (x) =
21. f (x) =
22. f (x) =
23. f (x) =
1
√
,
1 − cos x · cos 2x
1
√
,
1 − cos 2x · ch x
cos x
q
,
2
3
(1 − sin x)
1
√
,
1 + x − ln(e +x)
√
3
1 + sh 3x − 1
√
,
1 − cos x
√
√
3
sin x − 5 sin x
,
cos3 x
arcsin x
√
,
√
2 − 1 + cos x
1
√
√
,
3
cos 3x − 5 cos 5x
1
√
,
1 + sin 3x + cos 5x
1
,
cosec x − ctg x
x → 0.
x → 0.
x→
x → 0.
x → 0.
x→
16
π
.
2
x → 0.
x → 0.
x → π.
x → 0.
2
(1 − tg2 x)
24. f (x) = √
,
2 cos x − 1
π − 4 arcctg x
,
25. f (x) =
2
ln
x
√
√
5
sin x − 3 sin x
26. f (x) =
,
cos4 x
1 − tg πx
27. f (x) = 2
,
ln tg πx
π
.
2
x→
π
.
4
x → 1.
π
.
2
1
x→ .
4
x→
28. f (x) = √
1
x→
2 ,
3 − 2 cos x
1
,
29. f (x) =
1 − cos x · cos 2x · cos 3x
√
tg2 3 x
√
30. f (x) =
,
5
x2 sin x3
Íàéòè äëÿ ôóíêöèè
÷ëåí âèäà
1.
2.
3.
4.
f (x)
ïðè
π
.
6
x → π.
x → 0.
x → ∞
ãëàâíûé
Cxα
√
x
√
f (x) = √
√ ,
x+2−2 x+1+ x
√
2x − 4x2 + 1
√
f (x) =
,
3x − 9x2 − 4x + 1
√
x + x2 − x
√ ,
f (x) = √
x
3− x2
3
f (x) = (x + 3)2 sin e x −1 ,
ln (x + 2x )
√
,
5. f (x) = √
x+2− x−2
1
3
3
6. f (x) = x cos − cos
,
x
3x
1
1
7. f (x) = sin − tg ,
x
x
1
x → +∞.
x → +∞.
x → +∞.
x → ∞.
x → +∞.
x → ∞.
x → ∞.
1
4 x − 4 x+1
8. f (x) =
,
x
π
9. f (x) = x3 ln cos ,
x
x → ∞.
x → ∞.
17
1
· ln ch x,
x2
x+1
,
f (x) = x2 ln
x−2
r
r
3
4
4
3
f (x) = 1 + − 1 + ,
x
x
π
2
f (x) =
− arctg x ,
2
√
x2 + x + x
,
f (x) = √
2+1+x
x
r
2
4
4 1 + 7x + x
f (x) =
− 1,
1 − x3 + x4
10. f (x) = sin
x → ∞.
11.
x → ∞.
12.
13.
14.
15.
16. f (x) = π − arcctg x,
r
17. f (x) =
3
1 + arcsin
18. f (x) = x2 ch
x → ∞.
x → +∞.
x → −∞.
x → −∞.
x → −∞.
2
1 x +1
−
,
x x2 − 1
1
,
x
x → ∞.
x → ∞.
x4 + 4x3 − 2
x3
+
,
2x + 1
1 − 2x2
q
√
√
f (x) = x4 + x2 x4 + 1 − 2x4 − 1 ,
r
5
2
3 x + x + 1
f (x) =
ln sh x,
x2 − x − 1
√
x2 + 1
√
f (x) = √
,
4
x4 + 1 − 5 x5 + 1
√
√
f (x) = x 3 x + 1 − 4 x − 1 ,
√
f (x) = arcsin2 x2 + 1 + x ,
19. f (x) =
x → ∞.
20.
x → ∞.
21.
22.
23.
24.
18
x → +∞.
x → +∞.
x → +∞.
x → −∞.
25.
26.
27.
28.
√
ln (3 + 3 x )
√
ln ch x,
f (x) =
ln (6 + 6 x )
√
f (x) = 4x2 + 1 − 2x,
r
√
1
3
f (x) =
cos − 1
x + 1,
x
√
√
f (x) = x4 − x2 − 7 − x4 + x3 − 2 ,
x → +∞.
x → −∞.
x → +∞.
x → +∞.
1
1
· sin ,
x
x
√
30. f (x) = (x − ln ch x) x4 − 1 ,
x → +∞.
29. f (x) = arcctg
x → +∞.
4. Âûäåëåíèå ãëàâíîãî ÷ëåíà
ïîñëåäîâàòåëüíîñòè
4.1. Ïðèìåð ðåøåíèÿ çàäà÷è
Íàéòè äëÿ ïîñëåäîâàòåëüíîñòè
{xn }∞
n=1
ãëàâíûé ÷ëåí
C
.
nα
Ïóñòü ïîñëåäîâàòåëüíîñòü {xn }∞
n=1 çàäàíà ôîðìóëîé:
âèäà
xn =
q
√
n2
+1+n−
q
√
(4.1)
n2 − 1 + n.
Ïðåîáðàçóåì ôîðìóëó (4.1) ê âèäó
sr
xn =
√
n
1+
1
+1−
n2
19

sr
1−
1
+ 1 .
n2
Äàëåå, âîñïîëüçîâàâøèñü òàáëèöàìè 1 è 2, ïîñëåäîâàòåëüíî èìååì:
s
#
"s
√
1
1
1
1
xn = n
+1− 1− 2 +o
+1 =
1+ 2 +o
2n
n2
2n
n2
"s
s
#
√
1
1
1
1
= 2n
1+ 2 +o
− 1− 2 +o
=
2
4n
n
4n
n2
√
1 1
1
1
1
= 2n 1 +
+o
−
+o
+o
2
2
2
2 4n
n
4n
n2
1
1
1
1
1
−1 −
− 2 +o
+o − 2 +o
=
2
2
4n
n
4n
n2
√
1
1
1
1
1
√
= √ · √ +o
,
+o
= 2n
4n2
n2
n n
2 2 n n
1
1
òî åñòü xn ∼ √ · 3 .
2 2 n2
3
1
Ñëåäîâàòåëüíî, C = √ è α = .
2
2 2
4.2. Âàðèàíòû çàäàíèé. Íàéòè äëÿ
ïîñëåäîâàòåëüíîñòè
ãëàâíûé ÷ëåí âèäà
1. xn
2. xn
3. xn
4. xn
√
3
{xn }∞
n=1
C
nα
√
√
= n + 1 + 3 n − 1 − 2 3 n.
√
√
= n2 + 1 + 3 n3 − n − 2n.
√
√
√
= 4 n + 1 + 4 n − 1 − 2 4 n.
√
√
= 3 n3 + n2 − n + 3 n3 − n2 + n − 2n.
20
√
√
n2 + 2n + 3 n3 − 3n2 + 5n − 2n.
√
√
√
3
= 3 n2 + 1 + 3 n2 − 1 − 2 n2 .
√
√
= n2 + 2 + 3 n3 − 3n − 2n.
√
√
√
= 4 n2 + 1 + 4 n2 − 1 − 2 n .
√
√
= 3 n3 + n2 + n + 3 n3 − n2 + n − 2n.
√
√
= n2 + 2n + 3 n3 − 3n2 + 4n − 2n.
√
√
√
3
= 3 n2 + n + 3 n2 − n − 2 n2 .
√
√
= n2 − 2 + 3 n3 + 3n − 2n.
√
√
√
= 4 n2 + n + 4 n2 − n − 2 n .
√
√
= 3 n3 + n2 + n + 3 n3 − n2 − n − 2n.
√
√
= n2 + 2n + 3 n3 − 3n2 + 3n − 2n.
√
√
= 3 n3 + 1 + 3 n3 − 1 − 2n.
√
√
= n2 + 2n + 3 n3 − 3n2 − 2n.
√
√
√
4
= 4 n3 + 1 + 4 n3 − 1 − 2 n3 .
√
√
= 3 n3 + n2 − n + 3 n3 − n2 − n − 2n.
√
√
= n2 + 2n + 3 n3 − 3n2 + 2n − 2n.
√
√
= 3 n3 + n + 3 n3 − n − 2n.
√
√
= n2 − 2n + 3 n3 + 3n2 − 2n.
√
√
= 3 n3 + n2 + 1 + 3 n3 − n2 + 2 − 2n.
√
√
√
4
= 4 n3 + n + 4 n3 − n − 2 n3 .
5. xn =
6. xn
7. xn
8. xn
9. xn
10. xn
11. xn
12. xn
13. xn
14. xn
15. xn
16. xn
17. xn
18. xn
19. xn
20. xn
21. xn
22. xn
23. xn
24. xn
21
25. xn =
26. xn
27. xn
28. xn
29. xn
30. xn
√
3
√
3
n3 − n2 − 2n.
√
√
= n2 + 2n + 3 n3 − 3n2 + n − 2n.
√
√
= 3 n3 + n2 + n + 3 n3 − n2 + 1 − 2n.
√
√
√
4
= 4 n3 + n2 + 4 n3 − n2 − 2 n3 .
√
√
= n2 + 1 + 3 n3 − 2n − 2n.
√
√
= n2 + 2n − 3 n3 + 3n2 .
n3 + n2 +
22
Ñîäåðæàíèå
1. Âû÷èñëåíèå ïðåäåëà ïîñëåäîâàòåëüíîñòè . .
3
1.1. Ïðèìåð ðåøåíèÿ çàäà÷è . . . . . . . . . . . .
1.2. Âàðèàíòû çàäàíèé. Íàéòè ïðåäåë ïîñëåäîâàòåëüíîñòè èëè äîêàçàòü, ÷òî îí íå ñóùåñòâóåò
3
5
2. Âû÷èñëåíèå ïðåäåëà ôóíêöèè . . . . . . . .
7
2.1. Ïðèìåð ðåøåíèÿ çàäà÷è . . . . . . . . . . . .
2.2. Âàðèàíòû çàäàíèé. Íàéòè ïðåäåë ôóíêöèè
èëè äîêàçàòü, ÷òî îí íå ñóùåñòâóåò . . . . . .
9
3. Âûäåëåíèå ãëàâíîãî ÷ëåíà ôóíêöèè
9
. . . .
12
3.1. Ïðèìåð ðåøåíèÿ çàäà÷è . . . . . . . . . . . .
3.2. Âàðèàíòû çàäàíèé . . . . . . . . . . . . . . . .
Íàéòè äëÿ ôóíêöèè f (x) ïðè x → x0 ãëàâíûé
÷ëåí âèäà C (x − x0 )α . . . . . . . . . . . . . .
Íàéòè äëÿ ôóíêöèè f (x) ïðè x → x0 ãëàâíûé
C
. . . . . . . . . . . . . . .
÷ëåí âèäà
(x − x0 )α
Íàéòè äëÿ ôóíêöèè f (x) ïðè x → ∞ ãëàâíûé
÷ëåí âèäà Cxα . . . . . . . . . . . . . . . . . .
12
13
13
15
17
4. Âûäåëåíèå ãëàâíîãî ÷ëåíà ïîñëåäîâàòåëüíîñòè
. . . . . . . . . . . . . . . . . . . . . .
19
4.1. Ïðèìåð ðåøåíèÿ çàäà÷è . . . . . . . . . . . .
4.2. Âàðèàíòû çàäàíèé. Íàéòè äëÿ ïîñëåäîâàòåëüC
. . . . .
íîñòè {xn }∞
n=1 ãëàâíûé ÷ëåí âèäà
nα
19
23
20
Àëåêñàíäð Ïåòðîâè÷ Ãîðÿ÷åâ
Þðèé Íèêîëàåâè÷ Ãîðäååâ
Äìèòðèé Ñåðãååâè÷ Òåëÿêîâñêèé
Ìåòîäè÷åñêèå óêàçàíèÿ ïî òåìå:
Íàõîæäåíèå ïðåäåëîâ
Ïîä ðåäàêöèåé äîöåíòà À.Ï. Ãîðÿ÷åâà
Ðåäàêòîð Í.Â. Øóìàêîâà
Îðèãèíàë-ìàêåò èçãîòîâëåí À.Ï. Ãîðÿ÷åâûì
Ïîäïèñàíî â ïå÷àòü
. Ôîðìàò 60 × 841/16 .
Ó÷.-èçä. ë. 1,5. Ïå÷. ë. 1,5. Òèðàæ 2000 ýêç.
Èçä.  030 1. Çàêàç 
.
Ìîñêîâñêèé èíæåíåðíî-ôèçè÷åñêèé èíñòèòóò
(ãîñóäàðñòâåííûé óíèâåðñèòåò). Òèïîãðàôèÿ ÌÈÔÈ.
115409, Ìîñêâà, Êàøèðñêîå ø., 31