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1.

y = /"(x) (a, b), , x0 (a, b). 0, - > 0 5 () > 0 , x(Vx), | x 0 | < 5,

| /(x) A | < .

, , y = /"(x) 0

lim f (x) = A.

0

+ 0, - (0, 0 + 5)

| /(x) A+ | < .

- , - (0 5, 0)

| /(x) A- | < .

0 ( 0) 0 (x < 0), 0 0 ().

0 (x > 0), 0 + 0 ().

+ = - = , , /"(x) 0 .

 lim /(x) = lim /(x) = lim /(x) = /(x0) -

0 0 0

0.

0 /(x), /(x) .

0 :

1) /(x), Um /(x), /(x) 0, xlimx f (x) *■ f (x0);

2) , + -,

+ -;

7

3) , , , + = ~ - = + = - = ~. ( ~ + -.)

y = f(x) 0,

lim / (x) =

0

VN 5(N , (0 5, 0 + 5)| /(x)| > N.

y = a(x) 0,

lim (x) = 0.

0

(x)

() () ࠠ lim = 1,

w FW x^xo ß (x)

() i ß^) (() ~ ß()). , 0, sin x ~ x tg x ~ x; ln(1 + x) ~ x; arcsin x ~ x; e1 1 ~ x; 1 cos x ~ x2/2; (1 + x)k 1 ~ kx .. ( , ).

 lim U(x) = A, lim V(x) = B, :

x x x ^ x

0

1. xlx C = C( C const);

xx0

2. lim CU(x) = C A;

0

3. xm (U(x) + V(x)) = A + B;

4. xm (U(x)V(x)) = A ■ B

0

5. lim =A (B 0);

6. lim ln U(x) = ln lim U(x) = ln A, (A > 0);

x" yx x

0

7. lim a F = a - W = a*

8

, U(x) ) = 0, (U(x) V(x)) . -

0

lim Uixr , U(x) V(x)

xxQ (x)

= 0, - 0. 0 0 ,

U(x) V(x) 0. -, 0 . , 1. .

.

.. sin x ,

lim-= 1.

x0 x

:

,. sin kx , ,. tgx

lim -= k; lim -s = 1.

x0 x x0 x

lim (1 + x)1 = lim fl + ) = lim fl + ) = e, x^0v ' x^-{ x) x)

e = 2,71828...

. e lnx (ln x = log^x). ³ , logax.

䠠 1. :

x2 + 4x-5 lim -2-.

':

ϳ = 0 = 2, 15, :

x2 +4x-57

lim -:-= .

x^2 x2 -13

9

2. :

x2 + 4x - 5

lim--.

x1 x2 -1

/.'

ϳ = 0 = 1, ^ j. , ( >) = ( 1).

lim x2 +/x-5 = fl = lim (X + 5)(X-) = lim x5 = 3. x1 x2 -1 l. 0 ) x1 (x +1)( x -1) x1 x +1

3. :

,. sin ax

1im ;; a, b = const. x0 sin bx

/'

^ OJ. 1 .

.. a sin ax .. sin ax

lim - a lim -

lim sin ax = xO ax = xO ax = £

xOsin bx lim b sin bx b lim sin bx b'

xO bx xO bx

䠠 4. :

2x2 + x+1 lim -:-.

x-2x2 +x-1

/'

. ij 2 (

). , lim ~ = Jim = 0,

x- x2 x- x

:

,. 2x2 + x +1 ,. 2 + 1/x + 1/x2 ,

lim 5-- = lim -T-.-777 = 1-

x<*> 2x2 + x -1 x<*> 2 + 1/x -1/x2

10

5. :

1/ x

lim (1 + 2) ':

(1"). t = 2x . :

(1 + 2?x = (1-):

t = 2x x 0 t 0 : lim (1 + tf" = lim ((1 + t)Vf = e2.

䠠 6. :

lim (V2 + 2 -V2 - 2 :

, :

-4x^-2] + 4x^-2] (x2 +2) - (x2-2) lim ^-, , ->- = lim -^-'--

/2 + 4x^2 ^4+2 + 4-2

= lim ,-4 ,= = 0.

-^VX2^ +7-2

䠠 7. :

arctg 3x

lim --.

x0 6x

, arctg ~ 0. y = arctg . = tg y y 0, 0. ,

arctg y 111

lim -5 = lim = lim lim cos y = 1 1 = 1,

0 y0 tgy y0 sin y y0

. , arctg 3 ~ 3, 0,

.. arctg 3x .. 3x 3 1

lim-- = lim = = .

x0 6x x0 6x 6 2

11

1 ?

2 () .

3 () .

4 .

5 : ) ; ) I ; ) II ?

6 .

7 , .

sin x

8 : lim -= 1.

x0 x

9. : ) ln (1 + ) ~ ; ) ? 1 ~ ; ) (1 + ) 1 ~ ,

0.

: sin3x

1.1. ) Üm

cos(/ 2)

L2. )

1.3. ) lim

x0 Vx + 2 -2 ''

sin (x + ) - sin (x - ) _

arcsin (1 - 2x)

x1/2 4 x2 -1

sin4x

1.4. a) hmn-7=rT;

X0sl x + 1 - 1

,. 2x sin x

1.5. a) Imi --;

x0 cos x -1

,. 1 - COS 2

1.6. a) lim -j-;

1 - cos2x + tg2x

1.7. a) lim -

x0

2 xp2 cos2 x

1.8. a) Um

xsin x sin x

tg2^

6)

6) lim -

1 - 10n

) lim

10+1' sin (1 - x)

Hill

x- 1 x -1

6) lim (yjx2 + ax -x2 - axj

ln (1-

6) lim , x0 x

e-x -1

6) lim-;

x0 x

1-x

6) lim (1 - 4x) x ;

6) x- 0

e-3x -1

6) lim -;

x-~ x

12

˳: 1, .4, 18.

13

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