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Одесский национальный политехнический университет

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[НЕСОРТИРОВАННОЕ]

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b) y = tg3 (6x + 4x2 )

arcsin(4x)

y = (ln(2x))

.pdf

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'#8

f (x) 1, (x) .& x x

lim ( f (x)) ( x) = [1 ], #-

x x0

f (x) , (x) 0 .& x x

lim ( f (x)) ( x)

= [ 0 ], #-

x x0

f (x) 0, (x) 0 .& x x

lim ( f (x)) ( x)

= [00 ].

x x0

5% !$# 6 8 # ' ! !, 6. # "( 5 *#& +/(N/ ! $ A = lim f (x) ( x) ,

& /#N/, > ln A = lim (x) ln f (x) = . #" / /:

x x0

ln A = A = e lim ( f (x)) ( x) = e .

x x0

1. < #8 lim

x arctg(x)

$!’%$# %: & x 0 & /#N/ !$# 6 7 %

" . < #8 / . ' 6 57 "#

x arctg(x)

1+ x2 1

$#/ "#.

lim

= lim

1+ x2

= lim

x 0

3x2

x 0

3x2 (1+ x2 )

= lim

= 1 .

x 0 3x2 (1+ x2 )

x 0 3(1+ x2 )

2. < #8 lim

ln x

x 0 ln sin(x)

. < #8 / . ' 6 57 "#

$!’%$# %: & x 0 & /#N/ !$# 6 7 %

ln x

tg(x)

= '

$. & & /#N/ !$#-

$#/ "#.

lim

= lim

x 0 ln sin(x)

x 0

cos x

x 0

&%0#"

sin x

6 7

" , /( > $ $#' / . ' 6 57 "# $#/ "#.

lim tg(x) = lim

ln x

cos2 x

= 1 . lim

= 1.

x 0

x 0 ln sin(x)

3. < #8 lim

x 1 ln x

ln x

$!’%$# %: & x 0 & /#N/ !$# 6 7 [ ]. <! / 6. 57 * $#/- "#.

lim 1

x 1 ln x

x	= lim	1 x	= lim	1		= 1.
					1
		ln x
ln x	x 1		x 1

4. < #8 lim(cos(2x)) x2 .

x 0

& x 0 & /#N/ !$# 6 7 1 . *#& +/(N/ ! $ A = (cos(2x)) x2 , & /#N/

ln A =

ln cos(2x) .

ln cos(2x)

limln

A = lim

= %

$#' / . ' 6 57 "# $#/ "#

x 0

sin(2x) 2

tg(2x)

cos(2x)

= lim

= 2lim

= %

$#' / . ' 6 57 "# $#/-

x 0

"# = 2lim

cos2 (2x)

= 2.

x 0

ln lim A = 2, lim A = e 2

lim(cos(2x))

x 0

, . %.

									=1
	1	1	2		1		2	0	1	1 0 1			x	1		2	3	4
															3	2	9	8
	2	1 2			4		3	6	2		2	3 5	1
A =				B =		C =				D =			X = x2	F =	4	0	12 12
	4	1	4		2		1 1		0		9	2 0
													x3		1	6	3	0

1.< #8 G = AC 2(D + A) ;

2.$!’%$# /# & & ! % %CY = D ;

3. $!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

lim

2x3

+ 5x2 1

b) lim

x2 + x 12

, c) lim

x3 + 3x2

+ 2x

+ 4x + 3

x2 x

x + 8x

x 3

x 2 4 x

x 2

sin

d )

lim

, e) lim(3 2x)

1 x

x 0

x 1

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & # . -

							2	,	x < 2;
	-( (!# *+ " +( "2 ,: F(x) = 1 x							,	x < 2;
						x 2,			x - 2
7.	< #8 y/ 5% #6 (. ' +( "2 8:
a) y = 33			tg(6x);		y = arcsin(2x)
a) y = 33		x2	tg(6x);	b)	y = arcsin(2x)
					3x3 + 4
	y = e 3x 2 cos(6x4 )				y = (sin(x))
c)	y = e 3x 2 cos(6x4 )			d )	y = (sin(x))	tg ( x)
8.	<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y = 4x x2 ! 2 . & (

$ ! 66@ Ox .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 :

a) lim	tg(x)	; b) lim(x)sin( x)
	tg(5x)
x 2		x 0
)

										34
									=2
	1	3 3			1		1	0	2		1	1	1	x		3	2	5
	1	3 3			1		1	0	2		1	1	1	x		4	3	7
	2	1					3	4	6		2	1	0	1		4	3	7
A =	2	1	2	B =	0	C =	3	4	6	D =	2	1	0	X = x2	F =	1	1	2
	2 4						2	0	3		3	7	0			1	1	2
	2 4		1		1		2	0	3		3	7	0	x3		2	1	3
																2	1	3

1.#8 G = 2A C(D A) ;

2.$!’%$# /# & & ! % %YA = D ;

3.$!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

lim

1+ 4x x4

, b) lim

x +12

4 x

c) lim

5x2

+ 9x 44

x2 +

2x + 6

2x2

+ 5x 12

x + x + 5x2 + 3x4

x 4

1 cos(3x)

2x + 3

x+1

d )

lim

, e) lim

sin 2 (x)

x 0

x 2x +1

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & # . -

	2	,	x 2;
-( (!# *+ " +( "2 ,: F(x) = x
x 2,			x > 2

7. < #8 y/ 5% #6 (. ' +( "2 8

a) y = 2		3		; b) y = ln2 (sin(2x + 5))
	4x + 3

		x2 + x +1
c) y = (ecos( x) + 3)2		d )	y = (tg(x))x2

8.<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y = 8(4 + x2 ) ! 2 $ #-6- 2 6 @ x = 2 .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 :

a) lim x arctg(x) ;		b) lim(ctg(x))ln( x)
		1

x 0	x3	x 0

										35
									=3
3 2			2		1		2	1	1	1 2			3	x	6		2	5	8
3 2			2		1		2	1	1	1 2			3	x		11 3		7	11
	2	1	1				1	1	1		3	5	0	1		11 3		7	11
A =	2	1	1	B =	1	C =	1	1	1	D =	3	5	0	X = x2	F =	5	1	2	3
	2	1 2					3	5	6		7	6	2			5	1	2	3
	2	1 2			0		3	5	6		7	6	2	x3		1	1	3	5
																1	1	3	5

1.< #8 G = (3D A)C A ;

2.$!’%$# /# & & ! % % AY = D ;

3.$!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

lim

4x7 + 2x +1

, b)

lim

x +10

c) lim

6x + 8

5x7 1

2x2

x2 4

x +

x 3

x 2

cos(x) cos

(x)

d )

lim

, e) lim

sin

(x)

x 0

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & #


. -( (!# *+ " +( "2 ,:			2 + x,				x < 2;
			F (x) =		4	x2 ,	x - 2

7. < #8 y/ 5% #6 (. ' +( "2 8:
a) y = 5		ln(x + 5);	b) y = (			)ln( x)
	3x4 2
				6x
c) y = e x3 cos2 (x) d ) y =			3tg(2x)
			2x4 1

8.<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y = x2 4x + 5 ! 2 . - & ( $ ! 66@ Oy .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 :

a) lim	x sin(x)	;	b) lim(x)3x
	x3
x 0			x 0

									36
								=4
	1 1		2		1		6	2	0		0	2	6		x		1	0	1	3
	1 1		2		1		6	2	0		0	2	6		x		2	1	3	5
	1	5	1				3 1		2	D =			7	X	1		2	1	3	5
A =	1	5	1	B =	3	C =	3 1		2	D =	16 3		7	X	= x2	F =	3	1	4	7
	4	2 3					7	9	2			8	9				3	1	4	7
	4	2 3			2		7	9	2		2	8	9		x3		4	1	5	9
																	4	1	5	9

1.< #8 G = 3D A(C A) ;

2.$!’%$# /# & & ! % % AY = C ;

3.$!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

lim

3x2

, b) lim

x + 6

c) lim

5x + 6

+ 2x 4

x2 x 6

x2 9

x + 5x2

x 2

x 3

d )

lim arctg

(3x)

e) lim(2x

x 2

x 0

1 cos(5x)

x 2

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & #

		2
. -( (!# *+ " +( "2 ,: F(x) = 2x			1,	x 1;

	1 x,			x > 1

7. < #8 y/ 5% #6 (. ' +( "2 8:

a) y =			1 cos(3x)	ex ;	b) y = tg3 (sin(2x + 5))
c)	y =	arcsin(x)		d )	y = (ln(x))x2
			4x x3
8.	<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y = 4 x2 ! 2 . & (

$! 66@ Ox (x > 0) .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 :

a) lim ln(x) ;		b) lim(x)4+ln( x)
		3
x 3
	x	x 0

									37
								=5
	1 3		3		5		2	1	5	4 1 5		x	1		2	1	1
	1 3		3		5		2	1	5	4 1 5		x		1	4	1	1
	2	2 4					3	1			2 2 0	1		1	4	1	1
A =	2	2 4		B =	1	C =	3	1	9	D =	2 2 0	X = x2	F =	2	6	2	0
	2	2	4				4	1			1 0 2			2	6	2	0
	2	2	4		2		4	1	13		1 0 2	x3		3	8	3
														3	8	3	1

1.< #8 G = AC 2(D + C) ;

2.$!’%$# /# & & ! % %YC = D ;

3.$!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

lim

8x4 8x2 + 3

, b) lim

3 + 2x

x + 4

c) lim

6x2 +11x 6

4x4 +1

3x2

4x +1

x2 3x + 2

x 1

1 cos(4x)

x 1

x+2

d )

lim

1 cos(5x)

e) lim

x 0

x x + 3

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & #

	1,	x < 0;
. -( (!# *+ " +( "2 ,: F (x) =	2	x - 0
	(x 1) ,	x - 0

7. < #8 y/ 5% #6 (. ' +( "2 8:

a) y =

sin(3x)

;

b) y = 2 x2 ln(cos(4x))

+ tg(3x)

y = cos( 2x x

)

d ) y = (x)

<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y =

ln(x)

! 2 M (1,0) .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 :

	x2 tg 2 (x)	sin x
a) lim		; b) lim(ctg(x))
a) lim	sin2 (x)	; b) lim(ctg(x))
x 0	sin2 (x)	x 0

										38
									=6
2 3 0					1		6	1	1		1 1 2	x	1		0	4	1
2 3 0					1		6	1	1		1 1 2	x		2	1	5	6
	1	2	2				2	1	3	D =		1		2	1	5	6
A =	1	2	2	B =	1	C =	2	1	3	D =	1 1 3	X = x2	F =	3	1	9	7
	3	5	1				1	0	0					3	1	9	7
	3	5	1		2		1	0	0		1 3 0	x3		4	1	13	8
														4	1	13	8

1.< #8 G = 3D A(C + D) ;

2.$!’%$# /# & & ! % % AY = C ;

3.$!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

d )

lim

6x2 + 4x 5

, b) lim

3x2

+ 4x +1

c) lim

x +1

7x3 + x +1

+ x2

x +

x 1

x + 3

5 + 3x

x 1

+ 2

lim

1 cos(5x)

xsin(3x)

e) lim

x 0

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & #

. -( (!# *+ " +( "2 ,: F(x) =

+1,

x 0;

< #8 y/ 5% #6 (. ' +( "2 8:

ln(x),

x > 0

2 x

a) y = 4 cos(2x +1) + 4 e

; b) y = x

cos

3 + x

y = 2 x3 arcsin(4x) d )

y = (tg(2x))x2

<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y = e x ! 2 6 #-62 6 @

x .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 := ln 2

a) lim	ln(x)	;	b) lim(1 cos(x))ctg(x)

x 0 ctg(x)			x 0

										39
									=7
2 1			1		1		2	1	1	2 9 8		x	0		4	1	1
2 1			1		1		2	1	1	2 9 8		x		1	5	6	4
	3	4 2					1	3	0		1 12 12	1		1	5	6	4
A =	3	4 2		B =	3	C =	1	3	0	D =	1 12 12	X = x2	F =	1	9	7	3
	3	2	4				0	2	2		6 3 0			1	9	7	3
	3	2	4		2		0	2	2		6 3 0	x3		1	13	8	2
														1	13	8	2

1.< #8 G = 2D + A(D C) ;

2.$!’%$# /# & & ! % %YC = A;

3.$!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

lim

10x5 5x2 + 2

b) lim

x2 3x + 2

c) lim

x3 1000

2x5 + 3x 1

x3 2x2 +100x

x +

x 2

x +1

x 10

d )

cos(3x) 1

, e)

x 10

3x+1

lim

xtg(2x)

x 0

x x

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & #

	1		x < 1;
. -( (!# *+ " +( "2 ,:		,
	F (x) = x
	x,		x - 1

7.< #8 y/ 5% #6 (. ' +( "2 8:

8.<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y = x4 + 3x 2 ! 2 6 #-- 62 6 @ x = 2 .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 :

a) lim ln(x) ;		b) lim(x)4+ln( x)
		3
x 3
	x	x 0

										40
									=8
	2	1	1		1	4		3	7	3 3 2		x	1		1	1	2
	2	1	1		1	4		3	7	3 3 2		x		3	2	1	1
	0	2	1				1	1	2		1 1 2	1		3	2	1	1
A =	0	2	1	B =	1	C =	1	1	2	D =	1 1 2	X = x2	F =	4	3	2	1
	3	4	2				1	2	3		1 1 0			4	3	2	1
	3	4	2		2		1	2	3		1 1 0	x3		1	0	1	5
														1	0	1	5

1.< #8 G = 2AC (D + C) ;

2.$!’%$# /# & & ! % %CY = D ;

3.$!’%$# 6 6 /( AX = B , . .! 5( / / / K#(66#;

4.< #8 * /# & 2 F :

5.< #8 * 2 :

lim

5x3

6x + 2

, b)

lim

2x2 9x + 4

c) lim

+ 3x2 9x 2

+ 5

x 6

x 4

5 x x

x 2

d )

lim

cos(x) cos3 (x)

6x 7 3x+2

xsin(3x)

lim

x 0

6x + 4

6.65 +( "2 @ # . & &! 6 7, $ #8 " & $& !( ! %! , '#" & #

. -( (!# *+ " +( "2 ,: F(x) =

< #8 y/ 5% #6 (. ' +( "2 8:

y =

b) y = e

sin(x

) 5

;

cos

y =

arctg(x)

d )

y = ln5 (2x) + xsin x

ln(x2 + 5)

x2 +1,

x 1,

x< 1;

x- 1

8.<#. 6# & ! % % , # &/#5 *+ "# +( "2 , y = x x3 ! 2 6 #-62 - 6 @ x = 1 .

9.& 6 (@ 67 .! 5 / . #5% $ #8 #6 (. * 2 :

	e	x			1
a) lim	e		;	b) lim(x)
x x5				1 x
x x5				x 1

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