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Московский государственный машиностроительный университет "МАМИ"

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220

a)	z; = 6x 2 + 8x- 8 ; pewan ypasHeHHe z; = 0, H·axo.LJ.HM
x,	2				.			2
x,	=-, x, = -2 ; TOJlbKO O.LJ.Ha H3 JTHX TOlfe.K, x,							= -. ne)I(HT BHYTPH
	3 -						.	3 .
pe3Ka [-2; 2];		z	2)		22
pe3Ka [-2; 2];		z	-	=16-.
	.		2 ( 3	i	27
6) z2 ( -2) = z2 (2) = 32						-2)
s)	z2 ( ±2) = 32	-	HaH6onbwee, z2				22	HaHMeHbUJee
s)	z2 ( ±2) = 32	-	HaH6onbwee, z2				= 16- -	HaHMeHbUJee
						(3	27

Z2 Ha 3TOM 0Tpe3Ke.

ConocTaBn»» HaH60JlbUJee 11 HaHMeHbUJee 3HalfeHHH <j>yHKQHH

Ha yttacTKax AOB 11 ACB rpaHHQbl, npHXQ.LJ.HM K BbiBO)J.y: Ha sceH

HHUe AOBCA HaH6onbwee 3HatteHHe <j>yflKQHH z pasHo 32 (B

A 11 B), a HaHMeHbUJee- HYilfO (B TOttKe 0 ).

3) TaK KaK BHY1"PH 3a.LJ.aHHOH o6nacTH <j>yHKUHH z He HMeeT

HbiX TOtteK, TO ee HaH60J1bUJee 11 HaHMeHbUJee 3HatteHI-!H )J.OCTHrafOTCH

TOlJKaX, Jle)l(aiUHX Ha r.paHHQe o6naCTH:

z,6 =z(±2; 4)=32, z,... =z(O; 0}=0.

221

r;JIABA 5. BapuanTbl pac1:feTno-rpacjJH1:feCKHX pa6oT

PAClfETHO-rPA<I>HlfECKMI PAJiOTA flO TEME

«rPA<I>HKI13JIEMEHTAPHbiX <I>YHKUHH» (npHno:lKeH»e I)

3a.uaHHe I - 4. fiOCTJlOHTb rpa<jmK <jlyHKUHH. OnHcaTb OCHOBHbJe caoACTBa

38.ll8HHbiX <jlyHKUHi-1.

N!!

Bap.

<l>yHKUHll		N!!
		Bap.
I) y = jarctg { x -	I)j + 2	4.
I) y = jarctg { x -	I)j + 2

Z)y={./x+i, x~~l
x+l, x<-1
3) y!::: arccos(cos 3x)
4) y =secx
l)y=-log!. (x-2) +I		5.
3
2)y= {l/x,	lxl:>l	I
x,	lxl~+l

3)y =arcsin (sin 1)·

1-x
4)y=--
2x+ 1
1) y = 13 - arcctgxj	6.

2)y =lxl + j2x-lj

3)y =#sin x -1 + x

4)y= 2

2x +x

<l>yHKUHll

I) y = 21og113 (I- x)

rgx· X"'" /2

2) y =	tr	~
	(x- 2}(a-x), x>tr/2
3) y = x- ~1-lcosecxl
4)y =	I
4)y =	2

2x +3x+ I

l)y=. 3 2+1

(x + 1)

2)y= {larctgxl, lxl5.1 trl 4, lxl >I

3)y=1-lx2 -9j+1

I .

4)y= 2

,x +6x .

I)y = 3cosX- + 1

2) y=jx-lj·(x+ 5)

3)y=x·sinx

4) y = #2

x -10x+24

222
7.	l)y=2cos(1+ ~)-1		II.	I)y=2+3sin(~x- ~)
	l)y=2cos(1+ ~)-1			I)y=2+3sin(~x- ~)
	2) y= {	0,5x, X~ 0		2)y =		log!. x, x~l
	2) y= {	lx+II, x<O		2)y =			2
		lx+II, x<O			{-x2 -x+2, x<I
	3) y =.J1n cosx				{-x2 -x+2, x<I
	3) y =.J1n cosx			3) y = .Jtnsin x
		I		3) y = .Jtnsin x
		I		4) y = lcosecxl
	4)y= x2 -6x+8			4) y = lcosecxl
8.	1		12.				2
	I) y =-. 2.t-l + 3			I)y=			) +5
	2					(x+2)
						(x+2)
	2)y=(x2 -5x)·tx- 51			2)y=		~rctgx, x < 0
		x-5		2)y=	{ x+ti/2, x~O
		x-5
	3) y = .Jsin x
	3) y = .Jsin x			3) y =I+ .Jcosx + .J-cosx
				3) y =I+ .Jcosx + .J-cosx
	4) y=l2x-II						1
	x+3			4)y= xl-25x
				4)y= xl-25x
9.	I) y = lg .Jx - ') -I		13. I) Y =2+elx-ll
	. {J;+I,x~-1			2) y = {x			2
	2)y=	-(x+3)(x+I), x.<-1		2) y = {x			- 5x, x ~ 0
	2)y=
				.		arctgx, x < 0
				.		arctgx, x < 0

3)y=x+sinx

3) y = _:. + .JI-Jsecxl

4	1
	4)y=

4)y= 4x2 -1

10.I) y =I+ arcsin(x-3)

lxl-1

2)y=--

J2+xl 3)y=sin 2x

4) y = \fCOS X -I + _:.

14.1)y=(x-3)3 +5

2) y =JxJ+jx2 -41

3)y ~ x+ sin x.

4)y= :i -16

· 1	lxl
I)y=-tg-+ I
2	4
2) y =	arccosx, x ~ 0
	2	1i
{ -x		+x-- x<O
		2'

3) y = 2x + ~1-lsecxJ

4)y=-2-

x-9

1)y = 1+ 2arctg(I -~)

. {x2	-I,	JxJ ~ 1
2)y=		.
1-lxJ,		Jxl <I

J) y = _:. + .Jsin x- 1 2

4)y= xz -8x+l5

1) y = ~~- arctgxJ

· {sin2x, x S 0

2)y=	2		x>O
x-4x,
sinx	)		l
3) y=-- 4		y=-2-
x			x -4x+3
1) y =ltg(x- 2)+ 31
arctgx, x s 0
2)y= { 0,5x-l,			x>O

3) y =~-jx2 -11+2

'4) y = ) 1

x -9x

					223
19.			3	+1
			3	+1
l)y=- (x-2)2
.	{ log			1	x ~ 0
2) y=			--,
2) y=			2	x+l
	· -x, x<O
3)y=~-x2 +4x-4
4) y =xl -				I
4) y =xl -			6x2 + 8x
20.	1i
I) y = -+arccos(x + 1)
	2
	I,	x ~ 1
2) y = x2		,	Jxl < 1
	{0,		X S -J		•

3) y = ~log3 ( cosx)

4)y=l5x-31

1-x

21.I) y = -3arctg(x + 2,5)

2) y =12 -xJ +12 + xl

3)y= R 4)y=-1

. x-4x

22.1)y=ltog113 .fx'+l ....:21

2)y=	.JX +3, X ~ -3	'
2)y=	. .	'
	{ -(x+4)(x+3), x<-3	·

3) y = 3x- ~1-JcosecxJ

4)y= xl +2x2 +x

226

8. limn(~ -n) 9. lim(~-3n)			10, lim(~ -.J2,;)
II->-<	,_,, 2n + 3	,	11 ->xo	2n + 1	-
~ 1. !~,~ (sin J2n ~ I -sin J2n -I)		12	..{3;;'+1 .		n + 6
			limtg	sm	~
		.. II->"'		2n + 3	"4n + 3

I		.'	[ ·		2		]211
	3. Ilm I +				J311 2 + 11
		11-~r
BapuanT 3.
I		.	2n7 +11 5 cos11+3
	.	IIITI	411	6	·- 511	7	+I 0
		,_,~
		.	~411 4		+511 3		+~/2n5
3. I1m					4n + 7
		n-.-x:

14	.	( v2n+lr:::-	)J;;;J	. v2n+5~	- ....
14	. 1IITI	.j2,; +3	·.	151. 1m--===---==~
	11-u	.j2,; +3	·.	11->., ~_
		2. lim (311 + 4/(1- 4n)J
		II->'£	(211 + 1)8
+411 2		. (11+1)!(n+2)!
		4. 11m
		1,_."' ((11 + 3) !)2

211

311

5"... 2

4 ' 3

( 3 ~1

3 ~1

)

I + ... +.

+ ... +

I1111

211

1111

111111

'\111 ... +1-'i/11--1

11->,

11-u· J5211+1

+ 3

11->>0

lim(n-Jn 2 +2n+2)

lim( _42114

-2n2)

11->7

1,_,., 2n +3

10.

·rJI+2+3+ ... +211

2) 11

I' (. ~I . ~~)

IIITI

3n + I

1m Sin '\I 11 ...

+I -Sin '\In-

-I

n->r

11--+.()

1,_,,

<./3n4

+I +3

<.fn

2+4

13. !~"{)n'

211

+2 +3 ]

12.

J2n+3

.J3n+5

Jn2

+ 2 + 1

I1mtg

COS-==-

}.~~(1- 3~)

jJ;;;s

~-n

14.

11--+00 "n- + 1.

15.

lim {2"--::2- n

227

BapuauT 4.

I .

..2n9

+ 4n5 +n3

(2n + 1)9

I1m

11-+>:> 5n9

+ 22 cos n

411

+ 5)

,_,., (211 +3) (

~2n8 +4n5 +3+3n2 +4

4•

((n+1)!)3

I1m

11m.

11-+0J

3n2 +4n+778

ll->"'n!(n+2)!(n-1)!

211

(2. sn•l + 2 ;...))

J3 •7

+_998

hm ____ :_

11--+o<>

+ 211

IIITI -----:---:--

5211+2

11->Xl I + 7 + 72 + ... + 7"

limn 2 (<./n 3 +I-<./n3 -1)

lim j;;;l(J;;;2- .Jn+ 3)

.n-.'¥J

n-+·r-

4114

n5 + 2n4

1 )

lim(J

-2n]

. I1m

-n·

n-+tr

ll->>0 ( n2 +2n+2

11. lim (cos~·- cos~)

11-+oo

13.			.	I + 2vrn )(1-11)/(1-,f,;)				.
13.			ftm			I		14.,1IITI
		II-+"' [		3 + 2v n				II->"'
BapuanT 5.
l	.	I.	3n9 +4n3 +5·(-Ir					2.
	.	1m		4n	9	+n	7
		11--+"'		4n		+n	+12

12.lim 3 +(-1)" ll->><2+(-1)"

( n2	)(111 +2)/(11+1)
-'--	15•
n2 +I

3 '

arcsin--

n+4

. Vn· + 1 - n

11m ~

II->"'" n2 +I - n

. (2n + 3)4 + (311 + 1)4

I1111 -'------'---'-

ll--+r (2n + 1)4 +{3n + 2)4

,'I)'+

3211+1

. . </2n + 3 + 4n

+ 223

. (n + .

11.

3. )1m--,=====-~-

4. I1111

(11 + 2)n!

IITI----

II--+'.() .J3n4 + 4n3 + l

+ n2 + 5

11-:+oc

II-+-.:

2"+3 + 311

7lltl + 3ll-l

7. !~~( ~n + fn- Fn)

11--+"' 7 +72 + 73 + ... + 7"

6. lim----:---::----

8. lim n2 (<./n3 +l - ~n3 -I)

9. lim (

11 + J;;+J- J;,)

11-+oo

II--+X>

fn +2

228

I0.	hIll.	[(1+2+ ... +n)		- 11'1]	. •		1~
						11111 (cos v11 + I - cos v11)11
		n		2	II .
	n~1.					n-+-s'
		3 +sin 2n		J;+l +3				2
12.	lim						1'3. lim(	J;+I ) .r;.
			arctg~.--===-
	._., 4 +cos Sn			'<f2n 4	+ 3		11-+-<>	l+J;+i
			211 2				J;+l-~
14.	lim( nz + 2n + 2)				15.
	~~--."	11 2 + 3n + I				lim---==-
						11-+"' . n - </113		+ I
		.

BapnaHT 6.

	I) .		(8-nf+(8+nf
	•	1111	.	1			1
		II-"Y(8-n)·			-(8+11)
I.		.	1/'? 1	+	I	+	24/2	4	+	3
	3.	.	v ~n	+		+	v 11		+
~	3.	)IIll	1 ,----,----:				• ,..---;---:			4•
		n-u	\ln1	+1+3\t114 +1
						2			211
	6. )1111.		.J1 + 3 + 3				+ ... + 3			+ 2"
		11-+Y				3"+ 1

2.	.			(n + 2) 4
	hm				3		1
	n-+" (211 + 1)
						{vn
.	(	11	+	J)l(		2)1
				. 11-			•
11111

,_,,(11+2)!(11-l)!

+ 4)2

	. 12311+2 + 3. 311
5. 1Im		11 +2
11	-+"'211 +2·3

7	.	2 (J	(	4	+	I)	rs--;-)
7	•.11m11		11 11		+		-vn- -11

11-+:t:: .

lim(11 + </1- n3

)

9•

I1111. [11+(11+1)+(11+2)+ ... +211 - 11 J

,......~

II-> X

11 +I

10. !i~~(n1 +Fn -112)

II. !~~(sin111+1-sin</n)

11+2

{j4

+I)

n +1

12.

lun arctg <./

cos(n!)

13. hm

211

+ 3

(

•

11-H

5 311~ + 2

11-+"l

211 3 + 3

. (

~ ]"

,...."

2-11-.J4n2 +n+l

14.

ll-+7.

\t n3 +I +I ·

•

211- ,)4112 +I

I1111

,----,----:

11m

229

BapuauT 7.

(l-3n2 +411)3

(2n-1)4

~-~

I1m

11m

·,

11111 ,

r-;;-::

,---:;--:

11 ...."' (2n3- 3n + 7)2

II-+» (311 -1)(2\1111 -1)3

II-+X>

{J 11

4 + 2- v 11 3 +I

(n-2)!+(11+1)!

51. 1m

~~-+x4·311 - 1 +l

~~-+"'(11-2)!-(11+1)!

. ( 3 . 5

1+ 2 )

lim(</11

-11

+ n)

-+-+-+ ... +--

11 ...."'

n-+x:

lim 11

(~11

+1 -

fh1)

. [<.;;; + 1)2

.;;;]

+ 11 .J11

!~;

.;;;

11->oo

1O.

3 + 2

lim [ 1+ 5 + 9+ 13 + ... + (411- 3) _ 411 + 1] II. hm

sm2--

" ...."'

11+ I

II-+"' vn

+ 11

12.

lim (1 +

~ )"

13·

!~cos(l+(-1)")arcsin ~+Fn

n-t!Xl

3 + .J113 + 3

14.

lim

11 2 -

2n + l )"

IS.

lim

~-n

11,.."" (

112 + 411 + 2

II->»

:.r;;=:J: -Fn

•

BapuanT 8.

411

6 -5(-1)"n4

. (n -7)(611 + 5)(511- 2)

I1m __.,--:.--=...,----

2. I1m-'--___:_"'---'-'---_;_

n->«>

6n6

- 6n 2 + 1

"..."' (4n -1)(2n- 3)(1 - Sn)

~6114

+n3 +5 -Sn

(n+ 1)!+(n-2)!

. 4·311 +3 -3·211 -2

5. 1lm-~---

J../311 1 -

(11 +3)!

3"-

5. 2ll+l

n->"'

9n5 + 7n + 1

n-+a:

11->X>

1+ 5 +5 2 + ... + 5"

lim(.Jn

-3n+2-11)

6. I1m---:-----

n->oo

n->"' 1+ 3 + 3 + ... + 3"

230

lim( ~/(;;~-3-)-2

-1(n-

3)2

lim

11 J

11 2

n+2 )

n-Jr.

11 ->"' ( n

- n +1

10.

(1+2+3+ ... +n

- 3n

)

(

Iun

n+1

11 . 1un

cosvn+J3

-cosvn

11->X:

11-->:to

12. lim(1-4n)211... 1

)2n

13.

IIlim-+"' ( n2 + 5n- 1

11->:>:: 5- 4n

14.

limsin(.rn+2]cos(~) ,

I5. lim

j;;2;3 -n

,_,,

2n +I

2n + 5

,._.,., .rn+J - v n + 3

BapnanT 9.

1 I

5n6

+cos(n2

)

lim (3J;z + 2)2(3n- 2)

1111

•

6n 6

II-+.,

6n2 -

n +I 39

n-+r

n 4 + 29

3•

1+2+J+ ... +2n

41.

(n+2)n!

7"+1

I1111

r-:;--:

(n +I)!

5. hm

II-H "n4 + 3 +

<./4n6 + 5

IH-"

, ..... , 2. 7"-1- 311 +3

I + 5-I

+ 5-2

+ ... + 5_,

7. lim(Vn+5-Vn-5)

I1111----~--~-------

11-+>:

1+TI +.3-2 + ... +Til

11->Xl

r;:;-)

(

• (

2n + I

llmn("Vn

+3 -n) 9.

hm -·-. -n

10. hm

r;:;--; -v2n

II-+ X

ll-+0:• 4n + 3

h->:<c

v 2n - I

11.

!~1~ (sin .J2n + 3- sin.J2n- 3)

~. 2n+3

limtg

. sm

. 11_,.,

3n + I

v n + 3

13.

lim[ I -

2(3n

211

14.

lim(.J6n+3-Il.r,;;i

- 1r '

]

n~:t:

II-+.,

.J6n + 3

I5.

&+!-~

Iim ,.---:;--:

II-+~. "1/ n2 -

I - "1/ n2 + I

231

BapnanT 10.

3n7 +2n5 sin(n+I)+I3

2•

(n+4)6 (1-2n)3

I1m

2n7 - n6 + 10

1IITI

(n + 1)9

11 ..."'

II-+"'

~6n4

+.~3

+ <hn5 + 2n2

•

(n + 2)!(n + 3)!

5n +I

((n +4)!)2

II-+«J

II->X

lim 1+ 3 + 32 + ... + 3"

311+2

+ 7•-2

lim--;:===

n-+«>

3n+2

11-+.,

.j7211+l + 3

!~n413 (</n2

+2-</n2 -2)

8. ~~~(n-.Jn 2 +n+2)

6n4:.

2 )

(.JI+2+3+ ... +2n J

I1m

1111

-n

11->oo ( 6n2 + 1

.•

II-+"'

n +I

11. !~(sin.Jn2

+ 2 -sin.Jn2-2) 12.1im tg

J;;+2 -

cos ..;;:;+5 +3

~~~"'

3~24 I

Jr-;---::2.

... l?n"-1+

vn-+3

13.

lim

)4n

(

).[i.;';S

15.

311 -

1+ 2

14. lim

1- .

lim"~ -n

11->oo ( .J3n2 + 1 + 3

n->oo

2.J3n- I

II->'<>

BapnanT 11.

s .

(3n+1)

+smn

Ilm

+ n

+cos 4n

Itm _

___:__

_:__

n->.,

~~-+a:

(4n + 3)

(1- 4n)

</2n10

+ n5 + 3 - 2n2 +I

lim----:-----

~~-+"'

+ n + 578

((n -1)!)

•

(2. 311+1

2n-J )2

5. llm -'-~.----

4. Itm

n->., n!(n- 2)!(n + 1)!

n-+"'

2•32n+2

+ 2"

•../3·2211

+390

!~~ n

+ 2 -

<./n

6. Ilm

+ ... + 211

( <./n

- 2)

n->., 1+ 2 + 2

232

lim J;;+i(J;;+l- J;;+3)

9114

1 - 11J

lim ( .J

n-+x

n-+Y.)

( 11

+311

II. !~n;(cos.J11

+2 -cos.J11

-2)

/1-.-.:

n 2 + 11 + 2

11-

10.

Iun

12.

lim 3-(-1)" arcsin2--

I3.

lim (

I '---2../n J(l-n)/(1-,Jn l

n->><4+(-1)"

11+2

11->o: 5-2"11I

14.

lim(~)<,.2•31t<n+2l

I5.

~-11

11 2 + 2

lim

r:;----;: _

11-+f.

11->oc \111- + l.

BapuanT 12.

I •

211 8 + (-I)" 11 7 + 5

. (511 + 3)4

+ (511 + 1)4

I1111

4n8 + 11 4 + 18

Iun

+ (511 + 2)4

,_.,

n-o, (5n +I)4

~4n8 +3+n2 +623

(n+2)!+11!

.J4·32"-2 +3

3. Ilm

.J2n4 + 11 3 +I -112 + 9

41. lm

(11 + 3)n!

51. lm

211 +1+ 311 - 1

II-+•

"->"'

II-+.,

II->Y.

511+1 + 2211-1

!~~(~11 + ji;,- Frz)

5 +5 2

+5 3

+ ... + 5"

Iim-------

lim11 2 (~n3 +3-~n3

-3)

+Fn+2 _j;,J

lim(

11-+r.:

,__..,

Fn + 3

IO.

lim[(l + 2 + ... + 211)

-113 ]

II. lim(cos~11-I -cosif;,)

n-+x._

n-+~

3 + cos 711

Tn+2

+ 4

12.

lim

13.

lim(

.J211 +I J ,;;;

arctg

,r;::-;:--:

~~-+><

7+ cos 511

\1311 5 + I

II-+""

I+ .J211 +I

233

I4.

lim

112

+ 11 + 2

3112

.J211+l -.J211-I

15.

· ,. ....,.,

( 11 2 +411 + I )

Ilm

_:__-----=====--

ll->.:0

3 +I

BapuanT 13.

. (1-211)3 +(1+211)3

(11+3)4

I. I1m

2•

11m

n->oo (1- 211) 3 -(I+ 2n)3

11--->Xl (211 -I)3 ('-/n- 4)2

. ~-2~ 4

I' (n+2)!(n-2)!

I' ~53n-2 +33"

3. I1m

,--:-----

.~ . 1m

(11 + l)!(n -I)!

11--->0l

31 11 3 + 2 + 4\1114

+ 3

n->o<

II-+"' 2" + 2. 5"+

.Jt+3+3

+ ... +3

211

+3" 7

2(~

(

rs------:4)

I1m

3n+2 + 4 , 2n-l

- \111- -

,. ,.__..,

n-+oo_

8., !~~(11 + ~5- 113)

lim[l1 + (11+ I)+ (11 + 2) + ... + 311- n]-

ll-+.,

n +I

IO.

Iim(

113

-../n -n2 J

I I.

lim(sin~n+ 7 -sin if;,)

,__..,

n + 5

II-+"

~411 5

12.

-3

cos

(11!) I3.

· (J-2113 )"-

limarctg J

+ I

lim

4 -

2113

n-+oo

~ 2117

. "~"'

I .lim

~311

I .1im311-

911

-11+!

..}

~~-+., (~3113 + 2 + 8 J

,__.., 211- .J411 2+ 11 +I

BapnanT14.

I .

3n2

lim

(11 -I)4

I)2

Ilm

- n + sin(11 + 2)

· ,__.., 11

n-+"'('-ln

-I)2(2'-111-

. <h - n5 -

~2n2 + I

(11+1)!+(11+2)!+(11+3)!

_:__,==-~==-

Ilm~~~-~~~

n-+oo

~11 4

,__..,

(n + 4)!

234

lim

~hr+2

4"+3"

[

4/J_(n2_ )2'3]

6. hm

11m

. , ..... , 2 · 3"+1+I

, ..... ., .J1 + 2 + 4 + ;.. + 2"

,..... ,

limn3 (Jn2 +.[;;4;j -Jin)9. lim(

-~)

,..... .,

-~n.r)

,._."' 1+ 2 + ... + n

10. hm

II. limsin 3+(-1)" arcsin~

(.[;, +2

n-+x

4 n + 1

, ......,

7 -(-1)"

3n+1

,•

n-+oo

12. !~(---;;;:--)

-1

13.

lim(.fn+I-.Jn+2)sin(n+1)

15.1im~-~

, ..... .,

~l-n3

BapuanT 15.

. 2n8 -6(-1)"n4

-7

(n- J)(n + 5)(3n- 2)2

I. hm

----:---'-~-

Ilm

-1)(2n -1)(1-

, ..... .,

6n8

- n2

+ 1221

n-+ot (3n

5n)

•

~2n4

-n3 +5 -3n

(n+2)!+(n-2)!

• 2·9"+1-3·2"-2

I1111

+6n-1

4. 11m

(n+1)!

11m

n-+x{/4n7 -n5

n--.oo

n->oo

32n-

+7·2"+

I + 7 + 72 + ;.. + 7"

lim(n-.Jn2 +6n+2)

Ilm

, ......, 1+ 2 + 2 + ... + 2"

n-+a>

8. lim(~(n- 5)

- ~(n + 5)

)

lim

-~-

, .... "' n

)

n-+cc.

- 3n +1

n + 5

(

10.

(1+2+3+ ... +3n

)

(

~6)

+ 2

. 11m

cosvn -cosvn+o

n-+ot

n-+oo

12. lim ( I -

7.[;, Jln-1

n2 +3n

n-+«

6-7,J;;

•

n~( n2 +4n- I

. 235

. ..

(fn+2)

15.

lim

,---:--;;

14. hmsm

r cos

- n-.)

n-j;;2;8

, ......,

n + "n

+ 9

n-+» .Jn + 2 - "n + 3

BapuanT 16.

2n6

+3n+sinn

(2.[;; + 7) 2 (6n -1)

1. Ilm .

Ilm

-226n-3139

n-+oo

n + 29

n-+oo

·.

1+2+3+ ...+3n

(n+2)!

4·r+2

11m

,......--;::

5. hm

n-+"'...;n

+2+~9n6

,_...,(n+I)n!

•~-+"'3·7"- 2 +5"+4

I + 7-1

+ 7-2 + ... +

7-n

7. lim{J./n+9-</n-9)

I1m

+ T 2 + ... + T"

, ..... , 1+ T

,..... ,.,

,--:;-:

(

( 1-5n

r;:- )

)

8. limn(n-...;n- +lS)

n---

10, 11m

~+...;::m5 .

n-+oo

n-+"'

6n +I

, ..... ,

v 5n- 2

11.

lim(sin.J7n+ 4 -sin.J7n-4)

12.

·.

~. 4-n

hmtg

n-+oo

2n + 1

"n + 5

13.

lim[I+8(4n2 -l)-112 ] "

n-+ot:>

15.lim &+2-~

n-+oo .J,/- 5- .Jn2 + 5

BapuanT 17.
1.	.	·	7n7-2n5 +33
1.	Ilm		7				,.......,
	n-+«>6n		-2n4 +30·(-I)"				,.......,
3.	.	~3n8 + n3 + ~3n5		+ 2n	2	4.	lim (n+3)!(n+4)!
3.	hm	---=-------					n-+oo	((n + 5) !)'
	, ....."'	2n2 + 1222n + 1668					n-+oo	((n + 5) !)'
5.	liml+5+52 + ... +52"						.	22n+2 + 5"-1 .
5.	liml+5+52 + ... +52"					6.	hm --;::.===-
						6.	hm --;::.===-
	n-+«>		25"+1				,..... "'.J5zn+2· + 4

236 .

!~~ n413

(

\fn2 + 8- \fn2 - 8)

!~~(Jn 2

+ 4n +3 -n)

(5n

. . (.JI+2+3+ ... +4n

)

2 )

10.

-n

11m

n + 2

n--+-<

+ 1

~--+"'

II.

lim(sin~n2

-9 -sin)n2 +9) 12 limtg

J;+6 - 4

COS

J;+i +f;t·

II-+"<.

"--+"'

r:::-;-:-:

\fn2 +I

v8n

-1 +2 ·

13.

lim

"n2~+ 3 + 5JS11•

.[J;;+t

n - v n- + ()

n--+>O ( J:1+3 +I

lim

5~)

15.

J;T;4- n

.. n--+oo(

114"'

BapuanT 18.

4n8 -

+ I

'-(8n + 3)9

1. Ilm------

Ilin _..:.__...,..-.:..__...,..

114" 9n8 + (-I)"·n6 + 2

n--+>:> (n + 7)6 (3- n)3

~9n8 + n6 + 8- 9n 2+ 1

((n+l)!)3

I ill _

_;_:___;c...:;__

1111--------

.n.,.., n!(l'! + 2)!(n- 2)!

n--+.c

+ 457n- 8578

. (2·5"+1 -3"-3/

)2. 911

+ 30

5. hm -'---:-----.:.-

Ilin----::----

n--+-<

2 · 25"-

2+ 2"

~~--+"'1+3+32 + ... +3"

7. !~~ n

(\1n

9 - \1n

+ 9)

11--+oo

8. lim.Jn+5(..r;::ti -.Jn+7)

- n

9. I1111. (.J9n

+3 -

- n3 )

I1111

II-+>:>

3n + l

. n--+oo (

n2 + n +

II. !~ (cos .Jn 2

- 9 -cos .Jn2 + 9)

12: lim 6- (-1Y arcsin ,J;;+4

n->oo7-(-J)"

. n+l

r)(l-n)/(1-fnl

(

)(n2 +3lj(n+l)

13.

. 2-...;n

15.

. n--vn-+5

14. 11m

-.-

11111

,--,----:

n--+,., ( 7 -:- "n

II-+"'

2 + 6

II-+"' "n2 + 6 - n

237

BapuanT 19. I

+5n6

·sinn+15

. ·

(7n+3t+(7n+l)4

I1m

1111

3n8 + n6 + 98

II-+.,

n--+oo (7n + 1)4 + (7n + 6)4

{/8n8 +5-n2+77

(n+J)!+n!

. .J7-9"_1·TJ,

3. Inn

4. 1un

. 1111----

n--+"'.J6n4-nJ+l+n2+5

II-+«:

(n+4)n!

,,_., 211 +2 +311 -2

3. 511+2 + 22n-3

7. !~(J;, -~n+$n)

11--+oo 5 + 52 + 53 + ... + 5"

----=---:----

9. Jim(J;,- n- ;;;+I)

11--+<X>

(\In

-f./n

lim.n

11--+XO

j;, + 4

10.

lim[(l + 2

+ ... + 211)2

-n3

]

II.

lim(cos~n-9 -cosif;z)

11--+oo

n + 2

n--+O>

12.

I1111

2-cos5n

arctg-r===-

13. lim(

.rs;;+l

) ..r;.

.Jn+4 +I

n--+"'

3 +cosn

{/5n 5 + 2

II-+"' 3+.J5n+ 1

n2 +n

)211'

.·

.J5n+2-.J5n-2.

~~~~ (

+ 3n

+ I .

IJm

n -

{In3 + 2

' 11 -->"'

BapuanT 20.

l r

(l-6n)3 +(I +6n)3

(n + 2)4

·~~~~(I- 6n)3 -(I+ 6n)3

lim

ll--+"'(2vn-l)

(vn-

~4n3 + 1-6~

(n+3)!(n-3)!

. ~

1Jm

,~ .~

11-+00

3+7 +5\tn4 +3

ll-+"-'(n+2)!(n-2)!

II->>:> 3" +2•711 +2

.JI+3+3

+ ... +3

" +2"

2(~ (

rs-:)

IJm

-...; n- -IS

4 •311+1

11m n

n n

11--+oo

+ 5 •211-l

,..,...,

(

\IIr:::-~---:! )

un.

[n+(n:tl)+.(n+2)+.... +5n_

. 1m n +

9. 1

7 n-

n + 2

11--+oo

II->"-'

238

					173						2 ]		II. !~~ (sin .V11 ...., 7 -sin~)
IO.Iim(					+j;,						2 ]		II. !~~ (sin .V11 ...., 7 -sin~)
		, ....,._			n +9				-n
			.			~~-												•	(	3 -	411	3	)n·'+l
I2.			.			'I l'ln-				-	7					13.		•		3 -	411
I2.		hm arctg ~									cos3 (11!)					13.		hm		2-		1
		"->"'				6	6117			+I			.					, ......,		2-	411·
14.		lim(			{/113		+ 7				J2n		15. lim-.J_9_11_2r:-=2=n=+=1=-=3=11
		, ..... "			{/11 3 + 7 +I									, ..... , 211				-·.J	411 2·+ 2n +I
BapuauT 21.
I •		. .....:(_11_+~1)~(11_-_2....:.)~(2_n_+_3.:_)														2.		lim			(411 -It
I •		IIITI														2.		lim					--
		,......, (4n -I)(n + 3)(511- 2)																11-+"0 (Fn			-1)2(3];, -I)2
	.	.		(-./211+I +-./311+2)2													.	. (11-l)!+(n+2)!-(11+3)!
3	.	1111...:,.._______..:.......														4	.	IITI	~~_.....:___.:...._~~
3		1				411-3										4		I				(11 + 4)!
		II-+>:				411-3												II-+"'				(11 + 4)!
5.			·I-r"										6.	.	(	I		2		3 .			11 -1)
5.		lim--											6.	I1111		-2 +-2 +-2 + ... +-2-
		11 ..... " 1-3-11												II-+CC		11		11		11			11
7.		!~1! ( J;,- ~11 + J;+l)												8.	!~~(-J4n2					+ 11 + 1- 211)
		.		( 1+ 3 + 5 + ... + (211 -1)												~11 + 1)					.		( 211 2	+ 11	6_11 + 1)
																									3
9. I1111									11 + 1						---					10.	1IITI		311 -1		+--2
		11->"							11 + 1								2			'	11 ....."'		311 -1		1- 911
II.			limsin 4 +cosl1								·	Fn+6								12.	lim(l13 +3)n-n'
		11->>:			5	-	(	-	l)" Stn					.							, ......,		113 -2
						-		-					2311+5
I3.															11
I3.			Iim(J;+l--./11+2)costr
			n-+oro												2
					-JI12 + 311 + I--JI12 -3n -1
15.			lim						1rl				·
			, ..... ,					vi- 11-					+ n

239

BapuanT 22.

I.			. 31113 -4(-trn12 -8																2.	.	(11- 5)(11 + 4)(7n- 6)2
I.			hm		· 11	- I2n				1	+ I2I								2.	I1m
			, .......,		611	- I2n					+ I2I									, ..... ., (411 -1)(211- 4)(1- 8n)2
																			\
3		. .			~6114 -11 2 +9-n,													.	(11+3)!+(11-3)!						.	5·9"+2-5·2"-1
3			1												4. 11m										51. 1m	2		2
	. 1m				1	-11		3	-611-9						4. 11m					(n +I)!					51. 1m	2	+ 3 · 2"+	2
			"....."' 1./3n			-11			-611-9								, ..... .,			(n +I)!					, ..... ,.,	3 n+l	+ 3 · 2"+
			.	1+ 6 + 62 + ... + 6"															7.	lim(n- J11-i+ 811+ 8)
6. I1m																			7.	lim(n- J11-i+ 811+ 8)
			, ......, I+ 2 + 22+ ... + 2"																	,..... ,
		nlim(~(11-				-9)	2		-~(			11	+	9	)	2	)		9·	!~~ (				11 3 +I	n 2	)
8.		nlim(~(11-				-9)			-~(				+		)		)			!~~ (	11		2	-411+I- n+2
8.			+oo																				2	-411+I- n+2
			+oo
			.		(I +2+3+ ... +6n
10.			.		(I +2+3+ ... +6n										-n			)	11	.	(			r	~to)
10.			I1m					.	+ 2						-n				11	. 11m		coSvl1-cosvn+Iu
			, ......,			11			+ 2											,..... ,
12.			lim( I-9J;,l2n-l·																112					2
																			112					)_ "
													13.				lim(-			+ 6n
			, ......,		6 -9Fn								13.				lim(-			+ 6n
			, ......,		6 -9Fn												n-+oo		11 2 +11-I

		(rn+?l	2		-+))
14	• I..IITIStn		COS (n-	3	-+))
	n-+oo n + 2v 11		11		+3

BapuanT 23.

11- -J112	+ 12
15. lim	,---:-:::
,......, J;;+l - v n +		2 ·

·1	· .	3n7 +n·sin311+87
	11m-------
• n--+oo		411 1 + 5116 + 99
	•	1+2+3+ ... +7n
3. I1m		,-;--;::
	, ......, "114 + 2 + ~6n6 -9

2.	. (2-5f;,)2 (4n_-7)
2.	I1m.;__-~~..,.._~
	n->oo 11 2	+66711-31
.	(11+1)!	.	4·14"+ 2
4. 11m		5. ltm	2	1	+ 5"+	4
,......, (n- 3)n!		n->oo 2. 7 "-.			+ 5"+

1	2
6.lim l+T +T + ... +T"			7.lim(.Vn-ll--VI1+11)
,......, 1+I o-1 + Io-2		+ ... + 1o-n	,..... .,
8. limn(n-.Jn2	+8)	9. lim(n-~) IO. lim(~ +-Fn)
, ......,		,,..,	6n +I	, ..... .,	5n-	2

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