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Московский государственный машиностроительный университет "МАМИ"

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240

311

II.

lim (sin .J811 +I -sin .J811 -I)

12. limtg--sm .J

11 + 3

n->Y

,_,,.,

11 +I

13.

!~1~ [ 1- 6(511

14.

lin~( .J2n- 3)&:;4

-1r"

n->cll

.J2n

+ 5.

15.

lim

'JJ,;;l-~

,.......,-

n->>: .J11

2 + 11- 5 - "112

+ 11 + 5

BapuanT 24.

2119

211 5 + 3311 4

lim _(2_11_+_4....:.)_6(~1_-..:.:11~2),_.z

Ilm

·cos 611 + 3

(311 + 1)

, .... ., 511

211

n->"'

s .

lim (11-3)!(11-4)!

:\111-

+ 11

+ 'J 4n

3. Ilm

411 2

+ 82211 + 968

n-+"' ({11-5)!)

n->>:

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

22n+l - 5"-1

5. IIITI -------

J-im-;=====

,,....x

3. 25"-1

,....., .J3.

52n+1 + 2

!~~n

{~11

-l2-012)

n-+1-·

4 3

.8.

lim(n- .J112 + S11 +I)

lim(l12- 311•

+I)

I0.

!~~( 11

_ .J1 + 2 + 3 + ... + 211 )

n-+o<

3112 + 2

11+3

II. !~~(sin.J11 3

-9 -sin.J113 +9)

I' (j;T;3 + 5 J5n ..

J;+"3- 5

J2,;+i + .J3n

12. Inntg

cos

~11 2 + 2 .

.J11 2 + 3 + 1

, .... "'

~11 4 -I +I

n->oo

14. !~(1-

5~)

.jJii;i

15. lim 211 -~

"11- + 2-11

n-+oo

BapuanT 25.

1•	.	411 4 -·sin(11 +I)
	IIITI ___..:____:...
	n-+oo	11 4 + 112		+ 23
		~4118 -	11	6 + 2 - 511	2
3.1im....:._				_
	n-+<X>	311 2 + 411- 8
5.	.	(6·5"+2 +3"-2)2
	ltm ...;______...:._
	n-+o:>	3 ' 25n-l + 4II
7.	!~n2 (~113		-ll-~113 +ll)

241

	.		(311+1)9
2..IIITI		2	1	3
	, .... , (11		+ 2)"·(3- 2n)	3
4.	.		({11+3)!)3
4.	Inn-...:..:__:......:.....
	, .... ., 11!(11 + 2)!(11 +I)!

•.J4·9"+1+2

6.	Ilm	'
	n->x ) + 3 + 3- + ... + 3"
8.	!~1l.Ji1+3(~-.JI1+8)

	4				.	11 5			-11• + 1			3 )
9. I1111. (.J111 + 2-'11J			.	I	.	11 5			-11• + 1		-11	3 )
			.		IITI			2			-11
			I 0					2	+ 11 + 2
n-+oo	311 + 2	,		""""" ( 11					+ 11 + 2
11. lim.(cos.JI1 2		-12-cos.JI12							.	6+(-1)"			.	;;,
11. lim.(cos.JI1 2		-12-cos.JI12	+12)			12 .			I1111			arcsm-
,.....,									,.... ,.. 3+ (-1 )"					11 + I
.	r·)(l-n)/(1-../n)		. (	11	2	+ I	)(n2 +1)/(n+ll					.		3~
.	3 - v11	14.	.	11		+ I					15.	.	11 - vn- + 3
13. hm	----r	14.	ltm	-	-	+ 3					15.	hm	~
n-+oo ( 5 - "n		.	,.... ,	11	2	+ 3						, .... , "112		+ 9 - n

BapuauT26.

.			n 5 -7118 +5
I . I1111		8	+ n	6	sin(n!) + 38
n-+oo 6n
.	~-2n2 +7

3. I1111

n-'+<~>.J5n4-113+2+112+1

2.	.	o.-7n)4	·(7n+l)4
	IIITI ~-....:...,__:__....:._
	, .... ., (I +711)4		· (711 + 6)4
.	(n-3)!+n!		5		.	.Js-9"-2 +4
4. 1IITI				•	1	2"+3 +3"
		(11-4)n!			IITI
11-+"-					n-+x

.	4. 5"+3 -	22n-l		!~~ (J;, - ~11 + J8;;)
6. lt111 -----:-....,.----			7.	!~~ (J;, - ~11 + J8;;)
, ....., 5 +5 2 + 53		+ ... + 5"		lim (J;, _11- [;+2)
. 8. li111n 2	(~1-n3	+~l+n3 )	9.	lim (J;, _11- [;+2)
,.....,				n-+>:	J;, + 3

242

10.

lim[(l+ 2 + ... + 3n)

2 -n']

II. lil11(cos</n-12-cosif;;)

n---tf

n + 6

n-+-~

· 5- cos6n

.j;;;5 + n

.13. lim(

J%+1

)s.r;.

12 . .lim

arctg

II->~· 6 + v9n~+ 1

n-" 4 -cos 4n

{J n' + I

[

+ 3n

)'"'

IS.IimJ%+1-~

I 4. IIITI -,---

,,_."'

n-~n3

+ 7

n->>:

n" + 7n +I

BapuanT 27.

(l-9n)3 +(1+9n)3

(n + 4)4

I• II111 -'----'----'---'-

lim----

,.-H (I- 6n)3 - (I+ 6n)3

•

n->"' (2-Jn -I)(v' n- 7)

. ~9113 -7~

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

~/2·7 3 "-1 +4

3•

I1m

~ .r.--:

•

IITI

.1m

n-H

+ 3 + 6{J n

/1-t.O

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

n->o< 2" -7"+

.J);S+S2 + ... +52" +3"

•

2(~ ( 4 9)

~9)

I1111

6'311+1 +7 • 511-J

unn

nn+

-vn--'J

11->Y

11-t"'

3 ~3

[n+(n+1)+(11+2)+ ... +7n

8. Ilm(n+Vll1 -n)

-n

n- 2

n~-.,x.

,_.<J'".

10. !~~1~(113 +5-Jn -112)

II. !~1; {sin </2n -7 - sii1 €n)

n+l

( 3- 8n3 )"'+

~9n5

- S

12.

limarctg

~cos

(n!)

13.

lim

2- 8n·3

n->>:

3117

n->r

14.

!~~(~+I )

lll

._.,., 2n- .J4n

+ n + I.

IS.

~nJ

lim

.Jn2 - n

-n

BapuauT 28.

. (n + S)(n- 9)(6n +I)

I . I1m ...:____:_.;_____:_;:.______:_.

n-.oo (4n2 -1)(5n_;2)

3		.	(J8n +I + -f9n + 7 )2
		I
	• 1111 -0..----------"--
		/1-t«J		l-9n
		.	I - T"+	1		I	2
5.					• (
		h111	. __	1	6. hm	2	+-
		n .... oo	1- 3 n		n-+«)		n	2
						11

8. lim(.J4n2 +n+I-2n)

n->oo

I0. lim ( 2n2 + n + 6nJ + I)

._..., 3n -I I-9n 2

243

	2.		.			(7n-1)4
	2.		I1m	r;--			r::;-
			n-. ... (v6n -1)2 ('J7n -1)2
			.	(n -I)!- (n + 2)!+ (n + 3)!
	4• I1111 ..;__---'--'-----'----'--'-
			11->"'				(11 + 4)!
+	3			n -		I)	.	(	Fn -.Jn+rn+T		)
+	2	+ ... +--					7. hm		Fn -.Jn+rn+T
	2			n	2		n--+Y.
	n			n			n--+Y.
	9.		.	(1+3+5+ ... +(2n-l) 2n+l)
	9.		Iun				n+ I			---
		,,._..,					n+ I			2
	I I.		limsin 4 + cosn ·							J%+6
			11 -.., 5 _ (-))" Sill							23n+5

12. 'Jim ( 113 + 3 )n-n'				13. lim(Fn'+1--fn+2)cosJZ"n
	/1-t"l	113-2		11->X·				2
.	_ lim	( ~	)../n+J	.J	2 '	.J	2	-311-1
I4	_ lim	v n + 5 +I		IS. lim n	+, 3n +I -	n		-311-1
	n->oo	-Jn + 5 - I		"....~	~I - n3 + n

BapuanT 29.

. 6n9 -n·sin9n-99

(8 + 7-Jn)2 (9n- 5)

hm --=-----=----

. Iltn

+ 7n - 3 I9

~~->"'

74n9 + n6

+ 459

~~->«

3•

•

I+2+3+ ... +4n

(n+2)!

8·14"- 2

IIm

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

•

1101

(n- 3)11!

5• 11111

2 3

n->oo v' 114

+ 8 + </9n6 - 5

n->o::

n->x

3. 7 n-. -

6"+

lim

I+

+ 9

~2 + ... + 9-n

7. lim(</n-16-</n+l6)

II-'+'<>

I + Io-l

+ Io-2 + ... + I o-n

n->"l

10. lim (~l-9n

+v7nr::;-)

8. limn(n-vn·r:;--+'J)_:· 9. lim ( n---n -2)

n-.oo

n-..,

n +I

"_,"'

7n- 2

244

II: !~1~ (sin J9n + 8 -sin J9n- 8)

J4;;+j.

51--'n

12. lim tg

r--.;

11-+>0

5n + 1

vn + 3

13. lim[l-6(6112

-1}-112 ]

14. lim(.fi;z + 63)1%+4

11-+Y

11--+X

.fi;z + 5

15.

lini

~-~

r-:;-::

n-H J11 2 + 9n ~ 9- V11 2 + 9n + 9

BapnanT 30.

8n13 -6(-l)11 n12 -7

(n _:_ 5)(n + 9)(2 -7n)2

lun

7n 13 -2n7 +I

IIITI

(8n -!)(n- 4)(1 + 8'n) 2

11--u

~~--+•

3•

!:./9n 4

-n2 +7 -6n

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

. 6·9"+1 -7·5"-'

I1111

Z./J-3n 7 -Sn-5

11111

(n+l)! ·

Itm

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

,_.,

II-•"

11 -+"'

32"+1

+5·2'1+3

1+6+62 + ... +6"

lim(3n- J9n 2 + n + 3)

IJill

, .... •· I + I 0 + I 0

+ ... + I 0"

11-+~

8. ,!~1~(~(2n-1)

-~(2n+l)

)

9· ,!~1~

1r3 + 2

)

( n2 -5n+l- n+7

I 0.

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

II.

,!~~.(cos~- cosJn + 19)

IIITI

/1-+Y.

4n +I

12.

lim

1)7n-l

lim( 6n-n2

)s"

( 7

-6vn

13.

II-+>

6 I

n-+r.

3- n- n2

14.

lim sin

+ 1)

6n -J36n2 + 1

(

Fn+8 )cos (-

15. lim-==,---==

ll-+r.

2vn1 -n

n.-.y,

J;;+7 -Jn+ 2

~41;t ' .i

PACl.fETHO-rPA~Ml.fECKAH PAiiOTA·no TEME · _.; ;,,}:'/

'«llPE)J;EJI ~YHKQlflf» (npHJIO)f(eHHe 3).

BapuanT 1.

. .

x 2

- 4

2 + x ·

if; - I

lim

I - J x2 -

hm--

x-+2 x3 - 2x2 +X- 2

X--+-2

• x-+1

J;- I

4.1im

3 )

cos3x-cos5x

----

IIITI

xsin 7x

.HI

(

!-x

!-x·1

x--+O

. ../3+2sin4x-J3-2sin4x

sin3x

IIITI

7. IIITI--

,:f'<:

X-+0

.<-+.- sin5x

lim[l + tg 2

9. lim(l+ 2x2 ) 11(l-cosl.•l

(2J';)] ''

x-+0

X-+0

l/(sin2.uin5x)

I 0.

Jim

cosx .

II.

lim(sin 5x + sin8x)ctgx

x.-.o (

COS 2x )

x--+0

•,' I

12.

lim (x-Jrl2)tgx 13.

lim ln(l+3x+2x )

•

ln(l+2')

f,'

x-+n-12

•

14. hm ----=-

x-.

s m 6x

>-->-~ ln(l + 3')

15. lim arcsin 5xarctg 4x x-+o xsin 7xtg3x

BapuanT 2.

x3 -x2 -x+J

I' ..[;2;4_2 3

if; -2

l . IJm

-'J

• lm

lm --r--

x-+1

XJ +X -X

x--+0

x-+16 V X - 4

(

)

cosx-cosl

Jx2

+3x+1-1

Ilm

-----

5. 1lm

sin X- sin!

1lm------

:<-+-3

X+ 3

2 -9

X-+l

x-+O

sin 5x

lim

tg 5x

lim(cos5xyt8 '2x 9. lim

3/(.t-2)

10.

(

•

)1/(.t-5)

lim

X-+tr/2 tg3x

x-+0

X-+2 (

2 )

x-+S

sin I 0

246

II. lim(cosxcos3x)ctg2 7x 12.

lim(!- x)tg(Jrxl2)

x~o

x~l

13.

-e4x

14: lim 2' - 2

IS.

lim ln[tg(Sx + 1r 14)]

lim-----

In x

.• -.o

sin 3x

.•-•0 sin 4x- sin 5x

,_,I

BapnanT 3.

. x1 -6x

2 +llx-6'

2-~

x+1

I. Inn

X 2 -

11m

X 2

3. 11m .~

.t-+1

+ 2

_.... 7

.t->-l \f X+ 17 - 2

(

)

2x-sin5x

I1m

---

2 + 5x + 6

11m----

x->-2

x + 2

x-.o

x + tg4x

.jl+itg4x- Jl- 2tg4x 7

lim I- xJ

8. lim(l + 3x)Jism4.•

..t-+lsinJrx

x-+0

X-+0

Sll1

lim(2- X /2)'g(nl 4)

I0.

lim (ctg:x- 12)1/cos•

t-+2

X-+lt 12

II.

limx[sin(ll2x)+sin(ll3x)] 12.lim(ctgx-llsinx)

x~7

x-+0

~I+ 3x + 2x

-1

14.1im

tgx- tg2

e''"h- e•inlx .

tm------

15. lim----

sinSx

x-+2

In X -In 2

x->0

ln(l + 6x)

r-+0

BapuanT 4.

I .

x 3 +x- 2

·1.

rx+J3 -2J;+I

3 I' x2 -Tx

Itm

X 1

-X+ J

x2 -

. 1m

·r

x-+l

-X

x-+3

x-+1

"'X -I

(

x-4

)

sin(x3 )

I1111

x 2 - 5x + 4

3x + 2)

5. 11111

...... 1

3(x

....o sin 3 5x

sin 3xsin 5x

6. I1m

....o .J4 +sin 7xsin l3x- .J4- sin 7xsin 13x

tg1rx

)2/(x-3)

sm x

9. lim(2...: cos3x)

I1m--

8. 11m--

.... -2 X+ 2

x->3

(

sin 3

.t-+0

247

l 0.

lim(!+ sin 4x +sin 5x)11 ""6

II_.

lim x[tg(l 13x) + tg(l I 4x)]

x-+0

x-+~

I2.

lncos4x

14.

lim J'-2·'

I1m tg2xtg(x-1r

14)

13 . 11m

X-+tr/4

x-+0

In cos 5x

.t-+0

15.1im~--~

x-+o

arctg2 x

BapuanT 5.

x4 -1 .

~8+3x-x2 -2

x-8

l . Itm

2•

11m

X + x2

3. 1tm~

X-+- I X 3 + 3x + 4

. x-+O

.HS

~X -

(

·.

)

. 2-co_s3x-cos7x

I1m

3x2 -

5. I1111-------

•

X-+2

x 2 - x- 2

x->O

X Sin 9x

. .

r;::::.

r:::II

(

3...:.x

)4/sin Sx

'\jl~ -...;tg.

smx

ltm

-X2

ltm

x-+l .Jsin X -

-'-+" 1r 2

x->O 3 +X

lim(cosx+sin4x)11 x

IO. Iim(l+cos6x-cos2x)11 ""'sx

x-+0

l L limx2 [cos(21x)-cos(31x)]

I2. litjltg2Jrx·ctg3Jrx

X-+O?

x-+1

13.

limx(321x -I)

I4.

lim

2'inh -I

I5.

lim Jx·arctgSx

x-..,

x-+o ln(l + tg6x)

X-+O,

arcsin 6x

BapuauT 6.

x6 -1

J1+2;- 3

if; -I

Itm

1m--

x-+l x2 - 3x + 2

x-+4

"X - 2

x-+1 X- I

(

)

1-cos6x

. ___xt.:::g:...,.6=x

-----

11m

xsin7x

11m

•

.... 2

2-x 8-x3

.Ho

H 0 J-.Jcos5x

;

sin(x-Jrl3)

(

r::::- )3/x

Itm

I-2COSX

11m

cos....;

L2.X

X-+tr/J

x->0

248

lim [cos(l I x) + 2sin(l I x)r I0.

lim (I+ tg2x + tg3x)11 "i" 9

X~~

X~O

II. lim(l-cos 2 5x)ctg2 7x

12.

lim(! -Fx)tg(;rxl2)

.• ~o

.t-+1

arcsin(3x + 5x2 )

J'- 9

~1 +arcsin 5x -1

13 . I1111

•

14 • 11111--

15 . 11111

J1 + arctg4x -I

.Ho arctg(4x + 7x

)

.T-+ 2 x- 2

.T-+o

BapnanT 7.

-I

lim

X_:]

h111

---~ 2.

}jX+ I -1

lim---===

•->112 6x2

5x +I

.HI 2-

~X+ I 5

(

, -

)

cos3 5x -I

6• ·

tg(x- 2)

Iun

111n

x2-4

,_."

x(x- 2)"

x· - 3x + 2

....o I

-~cos 2 6x

tgx-sinx

[sin(.2+31x)]x

(5+3x) 'h

I1111

8. 111n

5 - 2x

11m

sin 2

x->0

>->0

· ,_,,

10.

lim (sinxY~'

II. limsin(x,- 4 )tg(;rx)

12. lim(l-x2)ctg;rx

T-+lfl2

T-+4

>-+1

1/ln(J-T)

. (2.,_1

·-

I'.Sill

I' -----

13 . 11111

( X' )

1111

In x

1111

x-+2

x-+1

.T-+0 5' + 7' - 2

HapuauT 8.

I .

x4 - 3x + 2

. ~-I

. x 2 -I

Ilm

x6 -I

11111

11111~

>-+1

x-+0

X-+1

~X -

(

x+2

x-4

)·

sin(x-I)

I1111

11m-_:__::_

x 2 -

5x + 4

3(x2- 3x + 2)

HI x 2

2x + 1

tg 7x

".L cos x - I

(

cos3

)4/(x-J)

6. I1111

(;;

1lm

I - tg2 X

1lm

x-+O }f4x + 8 - 2

>-+~r./ 4

>-+ 3

COS X

249

9.	lim(l + 2x2Yi(l-cosh) IO													,		(		COSX )li(sin2xsmh)
	.....o											.		hm			-----
														, ...o			cos2x
II. lim(sin5x +sin 8.t)ctgx															12.			lim (x- ;r I 2)tgx
		x-+0																>-+tr/2
13 .1i111 In(I+3x+2x2)												14. lim						ln(l+2')						15 .limarcsin5xarctg24x
		x-+o			sin 6x										H-"' In(I+ J')										HO		x sin 7xtg3x
BapuanT 9.
	.		x4 + x3 + 2x + 2											.		~+ 2									. if; -1
1. I1m					x 4 -1							2. 11m						X 3 + 8						3• 1un~
	x-+-1				x 4 -1								>->-2					X 3 + 8						.HI "X -1
4. limI -----6 J 5.														.		1Jcos7x-l								6.	.	Jx 2	+3x+1-1
4. limI -----6 J 5.														Ilm				xsin 5x						6.	1lm					.
	x->J		(	X -	3	X2 -			9					HO				xsin 5x							HO		sin 5x
·				· (1		)		·	·(1				)					(			5 )	3		5
·	.			· (1	+x	)		·	·(1		-x		)			.		(	cos		5 )		'< -•>			.		(		)'8 z.
7.	.		Sill		+x		-sill				-x					.			cos						9.	.		(	tgx	)'8 z.
7.	Ilm													8. 1lm .											9.	1lm			tgx
	HO tg(1 +X)- tg(l- X)															x-+5 COSX										Htr/4
.				( 1	t				2
.		lim		( 1	t			)l/sin		x		11.		lim				. .
10.		lim			+ .gx							11.		lim				tg4x · tg(;r I 4- 2x)
		x->0		1+Sill X										x->!f/8						.
12.		. .										13 .			.			Inc~s9x						14.	.	sinx-cosx
12.		I1msm 7;rx·ctg8;rx										13 .			11m			In cos 1Ox						14.	11m----
		x->2													x-+0			In cos 1Ox							X-+Hi4			In tgx
15		.		~cos6x-~cos2x
15	. Ilm							-
		.T->0					X
BapuanT 10.
1.	.		x4 +3x3 -2x-2									2.			.. .§+2';-5									3.	lim if; -1
1.	I1m				6			1				2.		11m				ll			2			3.	lim if; -1
	x-+1				X -			1						x->8				~X		-	2				x-+1 x-I
4	.		(	3 .				2			)		5		.				sin 7x
4	Ilm			-----									5		11m-===--
•	HI			)- Fx				1-if;						•	HO .J3x					+ 4 - 2

250

sin(3 + 2x)- sin(3- 2x)

1- sin(x I 2)

Iun

11111

f( -X

.t->0

X->8

(9- 2

8.lii11(J+tg2 (7J";))~'4x

(sin(5-41x})x

I 0

)tg(8xl6)

.-.o

9· 11111

. 5

11111

.--.~

.t->3

s111

ll.li111(I-\G)ctg(1ixl2)

12.lil112xsin(312')

13.1imsin 5 x-s~n 6x

.t=-+1

... -....1')

x ...... o

e7 x _ e ·'

14.

tgx- tg3

15.

· .

~I + arctg4x -1

IJm

h111-./

x-->3

In X- In 3

x->O

I -arcsin 6x - I

BapuauT 11.

x3+3x2

+2x

J;+2-~

. x2 -1

I1111 .

•

r;:--;

---:;r--

21lin r;--:-;

H-2

x -x-6

._.2 v4x+l-v5x-1

>->-1 -Vx+l

(

)

l-cos4x

tg(x-5)

Ilm

2x- x2

x 2 -

3x + 2

51. lm

2 - 25

x--.2

x-.o cos 2 3x- I

x-.s x

lim cos3x2-cosx

lim(cos(../h))6'·' 9. lim(tg(x+1il4))"tg2.•.

x->.or

x->0

I0.

lim (tg(x I 2))111 x-*' 2>

II. lim(sin x- sin 5x)ctg9x

x-->812

x-->0

12.

(-2-

)

-./I

+siri6x -1

14.

lim

Jnx

Iun .

sin 2 x

1-cosx

13 . 1tm-----

. ( x-l -1)

x-.o

x-.o ln(l+tg4x)

x->1

15 .

·sin9x

Ilm

x-.o ln[tg(5x + 1i I 4)]

BapuauT 12.

4+3x-x

if!+2x+I

·.

£-1

I . Itm

1tm

3• 1lm-;:r-

._._,

+x+2

H-I x+v2+x

-1

251

4	.	(	2			3 )			.	sin5xsin6x				6.	. _t_.!:gc._2=xt,;;g::::5=x=
	lt111	-----						5. hm		1-cos9x					11111	r--;-
•	x--.1	J~x		2	I-x3				.>->o						.>->0 1-vcos6x
	.	tg	5		8	.(	.		5 )4/(5-.>)		I'	(2			8 )"·•'
7.			x			I'	sm			9			-cos
	I1m--					. nn	--				. 1111				x
	X-->Jf sin 3x					x->5	sin X			. x->0 .

10. li111(2-x13r'8(8x 13 )

II.

limsin[(x-6)12]tg(1ixl12)

x-+3

x-46

12.

limx2 [1-cos2 (31x)]

13.

lim

arcsin 4x

14.

lim~

x->"'

x->0 ln(1 -

3x- X

)

x-> 0 5x - 2x

• 2 3

15. lim arcsm

x ·

x->0

arctg67X

BapuauT 13.

x3+ 2x +3

ifj;3;2 -1

if; -I

1. I1m

2. 11m

3• 1

+4x- x 2

x2 +X3

lm--;:r--

x->-1 5

x->0

x-->l

-I

4. lim(~-~)· 5.

lim sin

(4x)

6. lim

.[ti(JI 2) -.Jti;

x--.1

I- x · I- x

x-.o sin(x3)

x-> 112 ~sin(! I 2) - .Jsin x

.1im1~cos5x

.1il11(1-

x)3'sin4:

.lim(~-xl

)tg(8x!IO)

7 X->8 sm 2Jx

x->0

x->5

IO.·Iim(sinx)sinxl(x-sinx) '11. lim.x[tg(II3x)+tg(ll4x)] ·

x->0

x->~

12.

lim T tg(31T) 13.

lim arcsin(4x + 9xz)

14.

ln(1 + tg6x)

x-.~

x->0

arctg(7 X+ X 2 )

IJ m---'--=-...:...

3sin9x - 1

} '

15.

lim 2- 2x -

x->0

?x -J1x

252 BapuanT 14.

I .

. x3 - 3x + 2

2•

I1111

1lm

3..1lm -----;:--~

.HI X -4x+3

.HO.JJ+X+X2 -)

.HO)-~x+J

4•

(

)

sii1(5 + x)- sin(5- x)

•

sin

6xtg4x

I1111

-----

5•. 11111

lm ---;;::=====

l-x

1-x

tg(5+x)-tg(5-x)

.HI

<-+O

<-+ 0 1-Vcosllx

l+cos 2trx

1/stnJx

lim

8. lim

. .

lim[cos(8/x)+4sin(8/x)f

.<->1 12

tg22trX

x->O

( 6- X )

>->>:

10. lim(cosx/cos2x)111''" 6"inhl

I I. limx2 [cos(l/9x)-cos(l/x)]

x~O

x~z

3)llln(4-x)

_ 3

12.

lim (I +cos2trx)ctg 2trx 13. lim

14.

lim~

>->112

>->3 (

x->3 5' -5

2""6.<

_ 2sin5x

15.

lil11-----

x-+o ln(l- 2x)

BapuanT 15.

x4 -l8x2 +81

. ..fi+;-.[JT;i

I . I1111

2x 2 - 3x- 9

2. 11111

r;-:-

11111

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

x-+J

>-> 0

vi+ x-I

>->0

~6 x + 64-2

'( 3

)

cosx-cos4

•

tg7x

4. I1111 -----4

h111

1IITI

~ r:;

•-+I

1- x 3

I - x

.t->4

sin·x- sin 4

.<->o

v5x + l- v3

. . cos(x-2tr/3)

6. +5x

4/Jx

(

)"•

r:;

8. 11111 --

11111

cosx+sm 7x

. .

1111

x-u

V 3 - 2COS X

x->O (

6- X )

x-+O

I 0.

lim(J- 2x)181 ".12

II. lim(!:;- cos 3 5x)ctg2 4x

.t-+1

x-t>O

12.

limx2(1-co?(3/x))l3.1imln(l+tg8x)

ln(l+T)

X->>:

lm ____;_

• l·dill 7X

t-+-oo ln(l

+5X)

15. lim

arctg5x

253

>-+o {./tg2x·sin 2 9x

BapuanT 16.

·.

. .J3+x+x2 -.J9-2x+x2

x +4x+5

•

IITI

x 2 - 3x + 2

>-+-I 2x - 5x- 7

•-+2 .

4 .

( 3r::---;3-)

2sin x+sinx-:-l

c.osx-cos5x

X+'J)5 -x'

5. 11111-----

X-3sinx+l

x-+"162sin

X->->O

.t->0

Xfg9x

6• I1m. tg12x x.:..o)/4x-J +I

9. lim(x/8) ' ·<-B>

X-+8

.	cosx	8.	.	cosx 4/(6-x)
7. 11m		8.	1IITI	--
.t->tr/2{./(1-sin:\/		. .t->6 ( cos6 )

10. lim(l-sin3x.+sin5x)11"" 9 '

X-+0

11. limx(sin(l/x)-sin(3/x)] 12. lim(.Jx 2-x+l-l)·ctg;rx

X-+':(}

.t-+1

13 .

sin5x

lnx

.~-~

I1m.J

14. 1lm

>-+o

I+

7x + x

-I

HI sin(7:-

-1)

151. lm.

X-+0

BapuanT 17.

+4x

+4x+l

•

x-3

1-ctgx

I1m

1IITI

IITI

•

x-+-1

+ 5x + 2

HJ vx

2- ctgxctg 3x

-I _ 2

x-+" 14

lim(v(x+3)(x+4)+x)

3x+tg4x+sin3x

x-+-oo

1m---...::::___

__:_:_:::..:.:,

•

>-+o

x-sin6x

.: .J6+5sin4x-.J6-5sin4x

7 limsin3x 8.

lim(cos(.J8x'))2'·'

. x-+" sin 8x

•-+o

X-->0

lim(l+5x 2)1111 -cos 4xl

10.

lim(cosx)11""'2.t

x-+0

x-+2JI'

11. lim(cos6x-cos5x)ctg 27x· 12.

lim

ctg2x·ctg(x-;r/4)

X-+0

X-+Jr/4

254

13 ·

•

(56"

tgx- tg7

1·

2-T

I1mx

- -1) 14. lim

.t4X

1111

ln(l-5x)

nx-ln 7

.HO

BapnanT 18.

3 + 5x + 6

)' ..[;2;7- 4

tg2X- 3tgx + 2

Iun

-I

1111

x 2 - 5x + 6

x- 4tgx + 3

.H-1

.HJ

.Htrl4 tg

4. }~,._(x+-Jx2+2x+3)

5.limsin(6+7x)-sin(6-7x)

sin l5x

tg9x

(cos(x/2)-sin(x/2))x

1m _,___________,:___

cosx

Ix4o6-J3x2 +x+36

x4rrf2

8_.

lim(cos6x)

01S1 9x

[sin(9 +

(4/ x))]x

X40

9• 11111

--'-~__:__:.!_!_

x 4

-x;

sin9

10.

lim(l +cos9x-cos5x)11 ""hT

II. limsin((x-9)/2)·tg(Jrxll8)

x~o

x~9

12. limx2[1-cos2(6/x)']

arcsin(3.,Jx)

14.

lim

x _ 2x

x4.:

13 . 1un _

__:_.:____:__

.HO ln(l + 4.,Jx)

X40

15.

1- X

lim--·

lgx

Bapuan~ 19.

x4-x3 -2x+2 2

')' ~-3 3

sin 2x-2s,inx+1

Iun

X 2 -

3X + 2

1 r

-·

.H-8

2 + :V X

x-->tr/2

2sin

X+ Sill X- 3

lim x(N+2x+3 +x) 5. lim

s;n(3 -x)

X4->'

X 4J X -4x+3

xsin 9x

tg3x

(

)ctg4x

6. I1m J

-J

~sm

1m-- .

5-sin 2 4x~

5+sin 24x

x--.><tg5x

.<-+o

lim(2-x/9}'81 .orxiiB) 10. lim(cosx/cos2X)111 ";"sxsin?x)

x-+Q

x~o

I I.

12. lim(3--JIO-x)ctg3Jrx

1imx [cos(8/x)-cos(5/x)]

X-+00

X-tol

•

arctgx

x-3

\lt+2si~ 2 3x-l

•

141.

1111

•

lin

x--.oarcsin.3x·sin(x/2)

x--+JS'-125

x·arctg(2x)

x--+o

BapuauT 20.

+5x-6

-~-~

ctg3x-l

. Inn

2• 1rm

. 1m

x41

x _ 1

x4o

2ctg 2x _ ctgx -I

x4><14

(

r-;--:

r::--:;3 )

sin

(xl2)

5. 1lm---

vx· +5 +VJ3 -x'

x--+0 sin(2x3 )

x-+-.o

lim ~6+5tg6x-_ ~6-5tg6x 7.

lim l+cos3x

8. lim(4-x)sctg6.r

x--+o

tg5x

x--+~r!J

tg 2 6x

-<--> 0 4 + x

lim[cos(21 x) + 7sin(2/ x)Y I 0.

lim (I+ sin 3x +sin 7x)"•inQx

x--+~

•

U. lim x[tg(l I x) + tg(l 19x)J I 2.

lim (2x ·tgx -1r I cosx)

x-+oo

x-+~r/2

arctg(9x-8x2 )

14•

. ·Jn(l+sin8x)

•

2-JT -T

I1m

1rm

•

X--+O arcsin(5x + 6x

)

.T--+0

25'" 5X-I

sx -I 1-T

x--+0

BapuauT 21.

lim

x3 + X - 2-

X + I

Jim

tg1X+ tg 2X- tgx- I

-3x+2

2. hm --====--

x-.r

x--+-I.J4+X+X

-2

x--+3~r!4

tg3x -3tgx- 2

(

-2

)

. •

sin4xsin7x

X -X-

3 + 2X

5. 1lm------

x--+-oo

X--+0 COS5X + cos9x- 2

sin 8x

ctgx- I

+ tg

r)]Jix

. 1m4 1 r-;::-

1nn

6v x

X-+OI-{/cosl3x

X--+Jr/

:Vcos2x

.T--+O

lim(cosx +sin 8x) 11 x I0.

lim(sin 2x I sin 6) 111 T-J)

x-+0

x-+3

256

II.

lim(cos 3 5x-l)ctg27x

12.1im5"'·sin(3/5"')13.1imx·arcs;n 8x

x-+o

·'.... "'

x-+o

arctg 7x

•

2sin6t

14 .

smx-sm

15 .

·-

I!Ill

e• -

1!Ill

In( I+ tg7 x)

l-+O

e· ·'

.T-+0

BapuanT 22.

8x3 -I

2.Jx2 +x+l.-2-x

. ln 2x+lnx-t

I .

IIm

.lin

31.lin

+x-1

X 2

In 4 X - I

>-+lt22x

x-+0

x-+e

. .

tg(3

- tg(3- x)

lim

tg6x

}~~x(~+x) 5.

~·..Tosin(3+x)-sin(3-x)

Ho.J7x+5-J5

(

)4/(.t-9)

.JI+x·sin~-'\jcosa

lim

sm 9

lim(2-cos9xt'•

~~.To

tg2(x/2)

. .l'-+9 sinx

x-+0

10.

lim(tgx + tg6x)11 ''"Sx

11.

lim(sin 5x +sin 6x)ctg8x

x_,o

x-+0

.., .

(

Jl'X

)

(

· )l

ln[tg(9x+Jr/4)]

ln(cos7x)

1-·

1llntg

·Cg 7

Jl'X

• 1

11m~-___:;,

X-+0

ln(COS8X)

x-+1

Sin8X

arctg9x

15 . IIIll---;======'=====

~tg6x·si~ 3

BapuauT 23.

2x3 + 3x2 + 3x + 2

~I+ x + x2

~7 + 2x- x 2

I . Ilm

•

4 + 3x- X

x-+-1

•-+ 2

X -

lim

8sin 3 x-I

x-+lflo6sin 2 x-5sinx+l·

,r:;:-,)

. cos4

x -1

tg12x · tg5x

r---;:;--

xJim-+-~(X+ "\IY- x·

x-+0 1- '\"COS 7X

}_To 1- COSJ 8x

257

r.-:::-

)

(

)41son9.T.

lim

'\"Sm

lim

9. lim(x /9)

'<'·'> -

L.x- 8.

x-u/4 ~-I

x-+0

6 + 7X·

.T-+9

10.

lim(I- cos 3x +cos 5x)11sin'sx

II.

lim sin((x- I0)/ 2) · tg(JZ' xI 20)

X-+0

X-+10

12.

lim(l +cosJrx)·ctg 2JZ'x

13.

lim .J

arctg7x

x-+l

X-+O

I + 6x + 5x 2 - I

14.

lnx

15. ~~~

x-2

X-+1. Sl11 5

I IT

- 121

lim

. ( •-1 -I)

BapuauT 24.

-x

.J9+x-l

. .

ln 3 x+ln 2 x+Jlnx+3

2•

11m

. lin

1111--

x-+1 X

-1

x-+-8</15+2x+l

In 3 x+l

x-+llr

. 4.

lim(.Jx2 +3x+2+x) 5.1im

~g(x- 2)

..... '""'

x-+2 x -

5x + 6

:J5+3sin9x-J5-3sin9x

Jl+xsinx-cosx

tg8x

11111

x-+o

.....o

sin

(x/2)

8. .lim(cos(Ji1:;))5/x 9. lim(l- 4x2 )lt(l-cos9tJ

X-+0

10.lim(sinl2/sin3x) '<•-> II. lim(cosx-cos7x)ctg 2 6x

X-+4 X-+0

12. limx2 [1-cos 4 (9/x)]			13. limx(39'•-1)	14. lim tgx-tg8
......"'			x-+.,	, ....s In x - In 8
15.	z./t + arctg6x -I
	lim-"==~~=--
x-+0 .J1 -arcsin 8x -I
BapuanT 25.
.	x 6 -1	.	.Ji+;-~
1. Ilm----		2. hm --==-==
x-+-l 2x2- 5x...:. 7		x-'>0	~.h + X - <h - X

~ I

258

.	21nJx-21n2x+lnx-14. lim(2x+J4x2+x+9)
3. hm	I J •	I	x->-xo
,_,.	~ . n X -

tg(7+5x)-tg(7-5x)

lim

tg8x

Ilm

sin6x

r::-;-.

.....o

.....o 7 _ \19x2

+ 5x + 49

sin(trx 2 )

(

)"'n'2x

lim[sin(l1+(81x))]'

I1111

8. 11111

COS 9X

sin!!

,...., s in(tr x3 )

.<--+o

x-+-.:

I0.

lim (I+ sin 9x- sin 8x)''•inJx

II. lim x[sin(41 x) + sin(5 I x)]

J~O

X~~

12.

lim tg(Rx)ctg(7Rx)

arcsin(2\G)

14. ~i~~

x-•l

13 0 1un _

___;._....:._

5' -13'

~t-+O

In(I + 5\G)

15.

BapnanT 26.

x8 -l

. .

\G-1

ln 3 x+51nx+6

I. I1m

r;--

r;:-

..... _,x• -5x-6

3. 1

ln x+l

HI \/l+x-v2x

x-+ll•

X ...

x(J9x

+ 2x +3x)

5. lim

tg(

-x)

1·

~-~

lim

•

1111 ---;::::::::::::::--~=

5x + 4

4 x

</sinx- </sin 7

sin(x+RI3)

. [

(

r:-)]Jtson4x

(

-X 110

)•8 <ut2o>

1111

IITICOS'\/OX

1IITI

x-+-n!J I - 8cos3

x->0

x->10

10. lim(cos2xlcosx)111 xsonSx) II.

li111(1-cos3 6x)·ctg4x·ctg5x

x~O

x~o

arctgx3

12.

.tlim(!+_.Jr. cos3x)ctg27x

13: ~i~~ arcsin2 8x·sin(xl5)

14.

lim

lnx

15.

lim 11+3tg28x-1

.x-.1

sin(I0'-1 -\)

x-+0 X·arctg(xl7)

259

BapnaHT 27.

+5x-6

. ~8+3x-x2

-2

4cos 2 x-l

•

1un

. 1m

X6 -)

<fx2 +xJ

x->1

x->0

.t-uil2COS2 x-3COSX+I

•

( 3~ Jr::---:~3)

•

tg(x- 5)

4. I1m

Vx· + 71 + V!S8

- x·

11m--:"'-~--'-

..... ....,

x->S X2 - 6x

•

.J6 + 3tg7 x- .J6- 3tg7 x

•

sin(x -JZ" I 6)

I.· (

)Jctg7x

6. I1m

sin 11x

1tm

I- 4sin 2x

8. 1m

....o

.....': 16

~•->O

lim[cos(6/ x) + 5sin(61 x)Y I0.

lim(!+ tg3x + tg8x)

. '

X~~

x~O

II. lim(sin x +sin 6x).Jctgx · ctg2x

12.

lim (I+ cos 5x)ctg2 3x

x-tO

.t~Jr

x·arctg(7x-6x2 )

14•

ln(l+tg9x)

13 . I1m

arcsin 2 (5x +X2 )

1nn .

41 -I

x-+O

x-+0

3"'"

15 .

lim .JI +sin 4x - .Jt -sin 5x

x-+O

arctg7x

BapuanT 28.

2x3 - x -l

.JI - 2x -

JI + 2x

-==~---====

HI x2

-7X+ 6

x-+O </2- 3X -

</2 + 3X

(

3 )

4tg X+ 4tg X- tgx- I 4

Itm

tg3 X+ tgx + 2

\/'

$->lK/4

T-+-J<

sin7xtg8x

x·sin9x

. .

l-</sin3x

5. I1m

6. 1un

11111 -;====-

x->0cos5x+cos3x-2

x->O)-{/COS)4X

x-+KI 6 Jtg(3xl2)-)

8.1im[l+tg2 (13

$)t'' 9.lim(cosx+sin6x)11 ';"·'

.r-+0

X-+0

111121

10.lim(sin8xlsin4) ''- II. limx·[tg(41x)+tg(31x)]

x->112

,_...,

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