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Национальный минерально-сырьевой университет «Горный»

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( !

% , , a &

! ( f (x) , ! (

x	a	x = a ; !
& .
%. H limsin x							.
%. H limsin x						x2 + 3x	.
		x	1	2
& '.	7		x = 1						&
( , ,
limsin x					= sin ,1				= 1, 2 = 2.
limsin x			x2	+ 3x	= sin ,1			12 + 3 ,1	= 1, 2 = 2.
x	1	2				2

# (

00 , ,0 , , ,00 , 0 ,1 ,

! & & (. H . I, (.

I. (

lim Pn (x)

xa Qm (x)

) ,

0 ; !

( Pn (x)		Qm (x) n m
Pn	(x)	0	Qm (x)	0	,
	x	a		x a

, ( x a)k , k -

x = a , &. >

x = a .

%. H lim

x3 1

x 1 x2 3x

x = 1

& '. 8

. I

&. C

(

x1 = 2,

= 1

ax2 + bx + c = a( x x1 )( x x2 ) . 7

lim

x3 1

= lim

(x 1)

(x2 + x + 1)

= lim

x2 + x + 1

= 3.

3x + 2

( x 2)( x 1)

x 1

x 2

137

II. ( 0 ; ! '

		f (x)
? & (	lim		, f (x) g(x)	- &
		g(x)
	x a

& ! ( , ,

& , "

. & ( & ,

" ( .

%. 8 ( lim

1 2x2 1

x 0

& '. 8 x = 0 , ,

. Q ,

, , . . (

+ 1). 7

1 2x2

= lim (

1)(

+ 1).

1 2x2

lim

1 2x2

x 0

x2 ( 1 2x2 + 1)

H . 7 (

lim

1 2x2 1

= lim

2x2

= lim

= 1.

x 0 x2

( 1 2x2 + 1)

x 0 x2

(

1 2x2 + 1)

x 0 1

2x2

+ 1

%. 8 ( lim

3 + x

2 x 1

x 1

& '. >

. Q

, "

(

2)(

+ 2)(

+ 1)

(3 + x 4)(

+ 1)

3 + x

2 x

lim

3 + x

= lim

x 1

2 x 1

x 1

( 2 x 1)( 2 x + 1)( 3 + x + 2)

x 1

(2 x 1)( 3 + x + 2)

( x 1)(

+ 1)

2 x

+ 1

= lim

2 x

x 1 ( x 1)( 3 + x + 2)

x 1 (

3 + x + 2)

138

				;
&

I. ( ; ! + !
C	lim	Pm (x)	Pm (x) Qn (x) - m

	x Q (x) ,
		n

n , & xm ,

xn ( $ ), &,

(, & &$

& (.

%. H

lim

3x2

2x + 5

7x2

+ 3x 8

& '.

2x + 5

lim

= lim

(

= lim

8 #

x 7x2 + 3x 8

7 +

(

- & .

3 0 + 0

lim

3 8

7 + 0 0 7 .

7 +

%. H

lim

5x3

2x + 7

3x2

& '. 1 , ( " ,

						x	3	"	5	2			+			7		#					-	5		2		+		7 .
	5x	3	2x + 7					%										&					1									2
										x	2						x
											2															x2				x
lim				= lim				'										(	= lim x							x2				x
lim		3x2 8		= lim					2 "			8 #							= lim x				1				8				2 .
x		3x2 8		x					2 "			8 #								x			1				8				2 .
								x	%3							&							1		3						2
																											x		2
												x			2														2
									'			x				(							/								0
C & &																					5	, . .
C & &																					3	, . .
, x - & &$ .											>,																					– &
&$ , , ,				lim	5x3 2x + 7									= .
&$ , , ,				lim										= .
				x		3x2 8

1 (

139

%. H

lim

5x + 8

4x2 3

& '.

x . H

x2 .

=| x |

5x + 8

x%5

lim

(

= lim

(

4x2 3

x +

2 "

3 #

x +

| x | 4

3 .

% 4

(

7. . x

+ , x > 0 | x |= x.

5 +

x%5

5 +

5 + 0

lim

(

= lim

x (

= lim

4 0 2

| x | 4

x 4

& (

) . 1 ( x

2 ,

x < 0 | x |= x.

II. ( ; ! + !

lim

[

f (x) g(x)] , f (x)

g(x)

x +

x a

(

f (x)

g(x)

)

f (x) g(x)

& (

, f (x)

g(x)

+ ( & ),

x a

lim[ f (x) + g(x)] = + .

%. 8 ( xlim+ (x2 + 2x x).

& '. A . Q

(

x)(

+ x)

x2 + 2x

lim

( x2 + 2x x) = lim

x +

(

x2 + 2x + x)

= lim

x2 + 2x x2

= lim

x +

x2 + 2x + x

x + x2 + 2x + x

140

lim

= lim

=1.

x +

x2 + 2x + x

x +

2 "

2 #

x +

1+ 0 +1

%1+

+ x

+1&

x (

(

%. 8 ( lim%

& .

' x 2

(

& '. >

x3 8 = ( x 2)(x2 + 2x + 4) .

+ 2x + 4 12

+ 2x 8

= lim

lim %

x 2 ' x 2

8 (

x 2

= lim

(

)(

x + 4

)

= lim

x + 4

= 1 .

x 2

( x 2)(x2 + 2x + 4)

x2 + 2x + 4

# ! ; ,

x a + = +(x) 3 = 3(x) - & . H

lim +(x)

x a 3(x)

! & . > & " ( (:

1. # ! , , +(x) &

& , 3(x) ; ! $ +(x) = o(3(x)) ( : " +(x) 0 - 3(x) ").

2.# ! ( , , + ), ,

3(x) - & & , +(x) .

3.# ! & A 0 , ,

& +(x) 3(x) - .

4. # ! ", , &

H & (

! & .

. @ +(x) 3(x) , & x a ,

141

, ! ,

lim +(x) = 1.

x a 3(x)

H ! $ : +(x) 3(x) x a .

? ( ). &

& & , (

? ( ! ,). #

+1(x),+(x), 31(x), 3 (x) +1(x) +(x),31(x) 3(x)

& ,

lim +1 ( x)

x a 31 ( x)

x a,

=lim +(( x)) .

xa 3 x

? ( ! ,). C

, & +(x) 3(x)	& x	a ! &
, & , &		+(x) 3(x)			&
& ( & , +(x) 3(x) .
%. M +( x) = x 2 3( x) =				x
%. M +( x) = x 2 3( x) =			x2 4	x	2 + 0 .
lim	+( x)
lim	3( x) .
& '. I x 2+0	3( x) .

lim

x 2

= lim

x 2

= lim

x 2

2 2

= 0.

x 2+0

x2 4

x 2+0

x 2 x + 2

x 2+0

x + 2

2 + 2

E ,

x 2 + 0 +(x) = x 2

& &

, 3( x) =

x2 4.

C & &

& ( &

(& ), ! .

! & :

+(x)			0 , x a :
		x	a
1.	sin +(x) +(x) ,
2.	tg+(x) +(x) ,
3.	ln 1 + +(x)			)	+(x)				,
3.	(			)					,
4.	e+( x) 1 +(x) ,
5.	A+( x) 1 +(x) ln A ,
							+(x)
6.	n 1 + +(x) 1						+(x)			,
6.	n 1 + +(x) 1							n		,
7.	1 cos +(x)					[+(x)]				2
7.	1 cos +(x)							2		,
								2

142

8.arcsin +(x) +(x) ,

9.arctg+(x) +(x) .

%. H

lim

ln(1 + 3x2 )tg2x

(

)(

)

x 0

1 e

1 cos 4x

& '. 7 x

0 & 3x2

0 ,

" & !:

ln 1 + 3x2

)

3x2

tg2x

2x ,

(

1 ex = (ex 1) x,

1 cos 4x

(4x)2

= 8x2 .

3x2 2x

7 (

lim

4 .

x 0 ( x) ,8x2

& ' !

% 1. H lim

6x + 1

I$ . x x = 2 . 7

lim

3x 2

32 2

x3 6x + 1

6 , 2 +

x 2

% 2. H lim

x 2

2x2

x 3

x = 1 ,

& '. 8

x1 = 1, x2

= 2 . >, x2 x 2 = ( x + 1)( x 2).

L -

x1 = 1, x2 =

. M,

2x2 x 3 = 2( x + 1)%" x

&# .

( x + 1)( x 2)

2 (

7 lim

x 2

= lim

x 3

3 #

x 1

2( x +

1)% x

2% x

2 (

(

% 3. H lim

2x2

5x + 2

4x2 + 4x

x 2

& '. A

. 7 , "

, .

143

5x + 2 = 2( x

2)% x

(

x3 4x2 + 4x = x(x2 4x + 4) = x( x 2)2 .

5x + 2

2( x 2)% x

2x 1

lim

= lim

(

= lim

+ 4x

x( x

x( x 2)

x 2

( 2 - , (

( , & & ( – & &$ ). ( – & &$ . >,

lim

2x2 5x + 2

= .

x3 4x2 + 4x

x 2

% 4. H lim

1 + x

1 x

x 0

& '. A

(1 + x + 1 x ) :

(

)

lim

= lim

1 + x

1 x

= lim

1 + x

1 x

x 0

1 + x 1 x

x 0

( 1 + x

1 x )(

1 + x + 1 x )

x 0

(

1 + x

)2 (

1 x

)

1 + x

1 x

1 + x

1 x

= lim

1 + x +

1 x

1 + 1

= 1.

1 + x (1 x)

x 0

% 5. H lim

x2 + 1

x2 + 5

x 0

& '. A 0 . Q

(

1), (

+ 1), (

)

x2 + 1

x2 + 5

lim

x2 + 1

= lim

x2 + 5

( x2 + 5

), ( x2 + 5 +

), ( x2 + 1 + 1)

x 0

(x2 + 1 1), (

)

x2 , (

)

x2 + 5

= lim

0 + 5

x 0

(x2 + 5 5), ( x2 + 1 + 1)

x 0 x2 , ( x2 + 1 + 1)

0 + 1 + 1

144

% 6. H

lim

5x2 + 7x 3

2x3 8x2 + x

& '. A

. H

( $ ), x3

2 "

5 +

+ 7 x 3

lim

= lim

(

= lim

= 0 ,

= 0.

27 #

2 x3 8x2 + x

% 2

(

- & (& & &$ ).

% 7. H

lim

x2 7x + 8

+ 3x2

x2 11x

& '.

x x

+ 3x

7x + 8 + 3x

x, | x | 1

+ 3x

x x

lim

(

= lim

x2 11x + 3

x2 11x +

x2 11x + 3

7 x

, x < 0 , ,

x |= x.

+ 3

x, | x | 1

+ 3x2

x2 1

+ 3x2

lim

= lim

lim

(

x2 11x + 3

2 "

3 #

(

+ 3

+ 3 =

1 0 + 0

= lim

1 0 + 0

% 8. H xlim+ x(

x2 + 5

x2 5

& '. Q ,

" &, , x > 0 ,

. .

+ , , , | x |= x.

145

)(

)

(

x2 +5

x2 5

x2 +5

x2 5

+5 x2 +5

lim x

x2 +5

x2 5 = lim

= lim

)

(

)

x +5 +

+5 +

= lim

10x

= lim

10x

= lim

10x

5 #

x +

%1+

& +

| x| 1+

+| x| 1

+ 1

(

= lim

=5.

52 + 1

1 0 + 1 0

% 9. H lim

ln (1 2x) sin 3x2

arcsin

(2x) tgx

& '.

0 ,

3x2

0 ,

, ,

" & !:

ln (1 2x) 2x ,

sin 3x2

3x2 ,

arcsin2 2x (2x)2

= 4x2 ,

tgx x.

ln 1 2x

)

sin 3x2

lim

(

= lim

arcsin

2xtgx

, x

x 0

% 10. H lim

(

1)arctg3x

1 sin 2x

x 0

1 cos3 x

lim (

1)arctg3x = lim

(

1)arctg3x

& '.

1 sin 2x

1 cos x

1 + cos x + cos2 x

)

x 0

1 cos3 x

(

)(

> & !:

sin 2x

= x, arctg3x 3x, 1 cos x

1 sin 2x

. 7

(

1)arctg3x

lim

1 sin 2x

= lim

x ,3x

= 2.

(

)(

)

x 0

cos x 1+ cos x + cos

+ cos x + cos

cos0

cos 0 1

% 11. M &

+( x) = e2 x 1

3( x) = arctgx2

x 0 .

+(x)

& '. H lim 3 , &

x 0 (x)

lim	+(x)	= lim	e2 x 1	= lim	2x	= lim	2	= .
	3(x)		arctgx2		x2		x
x 0		x 0		x 0		x 0

E , arctgx2 - & & ,

e2 x 1.

146

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