пределы

¯®«ã®ªà¥áâ-®áâì â®çª¨ a, â® ¥áâì ¨-â¥à¢ « ¢¨¤

(a; a + ), > 0

(a

; a) .

•à¨¬¥à 1. •®ª § âì, çâ® xlim4 x2 = 16.

!

®ªà¥áâ-®áâì â®çª¨ 4, - ¯à¨¬¥à,2

¨-â¥à¢ « (3;5). „«ï ¢á¥å x 2 (3;5) ¨¬¥¥¬

jx + 4j < 9, á«¥¤®¢ ⥫ì-®, jx 16j

< 9 jx 4j. ’ ª ª ª -®ªà¥áâ-®áâì

â®çª¨ x = 4"(4 ;4 + ) -¥ ¤®«¦-

¢ë室¨âì § ¯à¥¤¥«ë (3;5), â® ¡¥àñ¬

j 4j

j

j

x!4

= min

1; 9 , ¨ ¨§ ¯à¥¤ë¤ãé¨å ®æ¥-®ª ¢¨¤-®, çâ® ¨§ -¥à ¢¥-á⢠0 <

< x

< á«¥¤ã¥â -¥à ¢¥-á⢮ x2

16

< ". ’ ª¨¬ ®¡à §®¬, lim x2 = 16.

x

! 1 ®¡®§- ç¥-¨¥:

lim f (x) = A

lim f (x) = A;

lim f (x) = A ,

x!+1

x! 1

x!1

¥á«¨

¤«ï «î¡®£® ¯®«®¦¨â¥«ì-®£® ç¨á«

" ( " > 0) áãé¥áâ¢ã¥â â ª®¥ ¯®«®-

i

h

8

i

2

x1. •à¥¤¥« äã-ªæ¨¨

x < C; x 2 E; jxj > C; x 2 E

¢ë¯®«-¥-® -¥à ¢¥-á⢮ jf (x) Aj < ".

h

•à¨¬¥à 2. •®ª § âì, çâ®

ilim

x cosx

= 0.

2

x!+1 x 10x+100

•¥è¥-¨¥. • áᬮâਬ «ãç x > 20, -

ª®â®à®¬ ¡ã¤¥¬ ¯à®¨§¢®¤¨âì ¤ «ì-

-¥©è¨¥ ®æ¥-ª¨. „«ï x > 20 ¨¬¥¥¬:

x2 10x + 100 > x2

x2

10x = x(x 10) >

;

2

á«¥¤®¢ ⥫ì-®,

x cosx

x

2

<

=

:

x2

10x + 100

x2=2

x

’ ª¨¬ ®¡à §®¬, ¥á«¨ C

20;

2

,

x > C á«¥¤ã¥â

= max

"

â® ¨§ -¥à ¢¥-áâ¢

x cosx

x cosx

< ";

lim

:

x2

10x + 100

â® ¥áâì

10x + 100 = 0

x!+1 x2

•à¨¬¥à 3. •®ª § âì, çâ® äã-ªæ¨ï f (x) = sin 1 -¥ ¨¬¥¥â ¯à¥¤¥« ¯à¨

x

x ! 0.

•¥è¥-¨¥. ‡ ¯¨è¥¬ á ¨á¯®«ì§®¢ -¨¥¬ ᨬ¢®«®¢ ã⢥ত¥-¨¥ \ç¨á«® A

-¥ ï¥âáï ¯à¥¤¥«®¬ äã-ªæ¨¨ f (x) ¯à¨ x ! a":

9"0 > 0 : 8 > 9x = x( ) : 0 < jx aj < ; x 2 E; jf (x ) Aj > "0:

…᫨ A = 0, â® ¢®§ì¬ñ¬ "0

=

1

¨ xk

=

1

, ⮣¤

2 k+ =2

2

8 > 09k 2 N : 0 < xk < ¨ jf (xk) 0j = jf (xkj = 1 > "0;

â ª¨¬ ®¡à §®¬, -ã«ì -¥ ¥áâì ¯à¥¤¥« f (x) = sin x1

¯à¨ x ! 0. …᫨ ¦¥ A 6= 0,

â® ¢®§ì¬ñ¬ "0

=

jAj

¨ xk =

1

. ’®£¤

2 k

2

…᫨ áãé¥áâ¢ãîâ lim f1(x) ¨ lim f2(x), â®

x!a

x!a

lim(f1(x)+ f2(x)) = lim f1(x) + lim f2(x);

x!a

x!a

x!a

xlim(!a

f1(x) f2(x)) = xlim!a f1(x)

xlim!a f2(x);

x1. •à¥¤¥« äã-ªæ¨¨

3

¥á«¨ xlim!a f2(x) 6= 0, â®

f1(x)

lim f1(x)

lim

=

x!a

;

x!a f2(x)

lim f2(x)

x!a

¥á«¨ ¢ -¥ª®â®à®© ¯à®ª®«®â®© ®ªà¥áâ-®á⨠â®çª¨ x = a ¨¬¥¥¬

f1(x) 6 g(x) 6 f2(x) ¨ lim f1(x) = lim f2(x) = A; â®

lim g(x) = A:

x!a

x!a

x!a

x2

1

xlim2(x2 1)

(

2)2

1

3

lim

=

!

=

=

=

1:

2x + 1

lim (x3

2x + 1)

2)3

2

2)+ 1

3

x

!

2 x3

(

(

x

!

2

t!t0

t!t0

t!t0

lim f (x(t)) = f lim x(t)

= f (a):

• ©â¨ x!4

1 + q

1 + sin

2

2

.

•à¨¬¥à 6.

lim ln

y1

=

x

; y2 = siny1; y3 = y22; y4 = 1 + y3;

2

y5 = p

; y6 = 1 + y5; y7 = lny6:

y4

x2. ‚ëç¨á«¥-¨¥ ¯à¥¤¥«

¢ á«ãç ¥ -¥®¯à¥¤¥«ñ--®áâ¨

5

…᫨ xlima f (x) = 0, f (x) 6= 0, â® xlima

1

= 1, ¨ ®¡à â-®, ¥á«¨ xlima f (x) =

f (x)

!

1

!

!

= 1, â® xlima

= 0.

f (x)

!

…᫨ xlim!a f1(x) = 1 ¨ xlim!a f2(x) = A, â® xlim(!a

f1(x)+ f2(x)) = 1.

…᫨ xlim!a f1(x) = +1 ¨ xlim!a f2(x) = +1, â® xlim(!a

f1(x)+ f2(x)) = +1.

…᫨ xlim!a f1(x) = 1 ¨ xlim!a f2(x) = A 6= 0, â® xlim(!a

f1(x) f2(x)) = 1.

…᫨ xlim!a f1(x) = A 6= 0, f2(x) 6= 0 ¨ xlim!a f2(x) = 0, â® xlim!a

f1(x)

= 1.

f2(x)

…᫨ xlima f1(x) = A ¨ xlima f2(x) = 1, â® xlima

f1(x)

= 0.

f2(x)

!

!ln(x3+4x+2)

!

•à¨¬¥à 7. • ©â¨ lim

.

ln(x10+x3+x2)

x!0

xlim0

ln(x3 + 4x + 2)

= xlim0 ln(x3

+4x +2)

1

= ln2 0 = 0:

ln(x10 + x3

+ x2)

lim ln(x10 + x3 + x2)

!

!

x!0

x

•à¨¬¥à 8. • ©â¨

lim

.

x

x! 1

2

x

= 2x x1. •à¨¬¥-塞 ã⢥à-

•¥è¥-¨¥. • áᬮâਬ ®¡à â-ãî ¢¥«¨ç¨-ã 2x

¦¤¥-¨¥ ® ¯à¥¤¥«¥ ¯à®¨§¢¥¤¥-¨ï ¨ ¯®«ãç ¥¬:

x! 1 x

x! 1 2x

x! 1

x

x! 1

lim

x

=

lim

2

x

1

=

lim 2

x

lim

1

= 0;

®âªã¤ á«¥¤ã¥â, çâ® x lim

x

= 1.

2x

! 1

p

+ x .

•à¨¬¥à 9. • ©â¨

lim

4x2 + 4x

•¥è¥-¨¥.

x!+1

’ॡã¥âáï

è¨å ¯à¨ x ! +1 äã-ªæ¨©, ¯®í⮬ã

x!+1

4 + 4

+

= +1

p

lim

x2

x

x

:

•à¨ ¢ëç¨á«¥-¨¨ ¯à¥¤¥«®¢

xlima

f (x)

;

xlima (f (x) g(x));

xlima (f (x) g(x))

g(x)

!

!

!

6

x2.

‚ëç¨á«¥-¨¥ ¯à¥¤¥« ¢ á«ãç ¥ -¥®¯à¥¤¥«ñ--®áâ¨

0

) lim f (x) = lim g(x) = 0 (ᨬ¢®«¨ç¥áª¨ ®¡®§- ç ¥âáï

);

x!a

x!a

0

(ᨬ¢®«¨ç¥áª¨ ®¡®§- ç ¥âáï

1

2. lim (f (x)

g(x))!:

¡) lim f (x) = lim g(x) =

1

).

x!a

x a

1

x!a

xlima f (x) = 0, xlima g(x) = 1 (ᨬ¢®«¨ç¥áª¨ ®¡®§- ç ¥âáï [0 1]).

!

!

3.

xlima (f (x) g(x)) :

!

xlima f (x) = xlima g(x) = 1 (ᨬ¢®«¨ç¥áª¨ ®¡®§- ç ¥âáï [1 1]).

!

!

lim

sinx

= 1;

lim

ex 1

= 1:

x

x!0 x

x!0

1 cosx = 2sin2

x

x2

sinx

;

tgx =

x;

2

2

cosx

arcsinx sin(arcsinx) = x;

ln(1+ x) eln(1+x) 1 = x:

x2. ‚ëç¨á«¥-¨¥ ¯à¥¤¥«

¢ á«ãç ¥ -¥®¯à¥¤¥«ñ--®áâ¨

7

•ª¢¨¢ «¥-â-®á⨠¯à¨ x ! 0

sinx x x2

1 cosx

2

tgx x

arcsinx x

arctgx

x x

a

xe

1 x

1 x lna

ln(1+ x) xx

+ x)

lna

loga(1m

(1+ x)

1 mx

•à¨¬¥à 1. ‚ëç¨á«¨âì ¯à¥¤¥«

lim

anxn + an 1xn 1 + : : : + a1x + a0

(

m >

1

; n >

1

; a b

m 6= 0)

:

x!+1 bmxm + bm 1xm 1 + : : : + b1x + b0

n

§- ¬¥- â¥«ì ¤à®¡¨ -

xm.

11 . • §¤¥«¨¬ ç¨á«¨â¥«ì ¨

•¥è¥-¨¥.

ˆ¬¥¥¬ -¥®¯à¥¤¥«ñ--®áâì ¢¨¤

lim

anxn + an 1xn 1 + : : : + a1x + a0

=

x!+1 bmxm + bm 1xm 1 + : : : + b1x + b0

a1

a0

a

xn m + a

n 1

xn 1 m + : : : +

+

x

m 1

m

=

lim

n

x

=

bm 1

b1

b0

x!+1

bm +

+ : : : +

+

x

xm 1

xm

lim

a xn m + a

xn 1 m + : : : +

a1

+

a0

=

m 1

m

x

!

1

n

bm 1

b1

x

b0

x

=

+

n 1

x!+1

m +

x

+

+ xm

1 + xm

lim

b

: : :

1

lim

a

xn

m

a

xn 1

m

: : :

a1

a0

:

n 1

+ xm 1

+ xm

= bm x!+1

n

+

+

Žâáî¤ ¯®«ãç ¥¬, çâ®

=

8 bm

x +

m

¯à¨ m > n;

anxn + : : : + a0

<

0

lim

an

¯à¨ m > n;

+ : : : + b0

! 1 bmx

1 ¯à¨ m > n:

:

•à¨¬¥à 2. ‚ëç¨á«¨âì ¯à¥¤¥«

lim

5x32+2x2 4

.

x!+1

6x +8x+9