PowerPoint .
,
(i)
(ii)
(iii)
10
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0.2
0.105
100
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0.01005
1000
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0.000001
0.002
0.0010005
n
1. (x), (x), (x), (x). f (x), , (x) ~ (x), (x) ~ (x),
.

5. x0, tgx – sinxx?
: : x0, f (x) = O(x), g(x) = O(x), f (x) · g(x) = O(x+), , , 0.
tgx – sinx = tgx(1– cosx)
tgx ~ x , 1– cosx = O(x2).
tgx – sinx = tgx(1– cosx ) = O(x3).
.
,
= 1
f (x) g(x) = O(x),
. , .
, f (x), f (x).


f (x),
, , x0.
f (x)x0.
"0<|xx0|< "" |xx0|< ".
1.
:
2.
a,
f (x) x = 0.
f (x)x=0, f (0–0)=f (0)=f (0+0)
, a=3.
= a
3.
f (x)C(a, b).
C(a, b)(a, b).
f (x)C[a, b].
, uu0u1,
, u1 = u0+ u
y = f (x) f (x0) = f (x0 + x) f (x0)
yx0x().
f (x)U(x0),
xU(x0),
3.y=f (x)U(x0).
x = xx00, y=f (x0+x)f(x0)0
f (x)x0.
(2) f (x) ·g(x)x0.
(3) g(x0)0,
3. y=f [(x)] y=f (u), u=(x).
u=(x)x0,
u0=(x0),
,
:
y=f [(x)]x0, >0, >0, |x–x0|< , | f [(x)] –f [(x0)]|<. .
>0, y=f (u)u0,
> 0, |u–u0|<, | f (u) – f (u0)|< .
u=(x)x0.
> 0,
>0, |x–x0|<, |u–u0|= |(x) – (u0)|< .

y=f [(x)]x0.
. lim[(x)] =A. y=f (u) u=A,
limf [(x)] = f [lim(x)]

4. y =f (x)I(),
x=f –1(y)() .
5. y =f (x)x0, f (x0)>0 (<0), U(x0), x U(x0), f (x)>0 (<0).
6. (1) .
y=[f (x)]g(x), f (x)>0.
y=elny, y>0, [f (x)]gx = eg(x) ·lnf (x),
,

6.
7.
8.
9.
y
x
0
1
10.
y
0
1
x
1
limf (x)=1, limg(x)= , lim[f (x)]g(x) “ 1 ”.
limf (x)=0, limg(x)= 0, lim[f (x)]g(x) “ 00 ”.
“ 1 ”, “00 ”“0 ”, , 1.
limf (x)= , limg(x)= 0, lim[f (x)]g(x) “0 ”.
11.

(1) f (x) x0;
(2) f (x) x0;
1.
:
.
.
x=0f (x).
, :
(0, +), f (x) = sinx, .
x = 0.


x
o
y
–2
–1
0
1
1
y
x
1. (), f (x)C[a, b], f (x)[a, b]. f (a) f (b)<0.
x0(a, b), f (x0) = 0.
.
,
1.
0
a
b
x
y
A
B
x0
x0
x0
2. (), f (x)C[a, b], f (a) f (b),
f (a) f (b)c, x0(a, b), f (x0) = c.
.
x0
C
0
b
x
y
: F(x) = f (x)–c.
F(x)[a, b],
F (a) F (b) = (f (a)–c)(f (b)–c) < 0
, x0(a, b), F(x0) = 0., f (x0) = c.
1. ln(1+ex)=2x1.
f (x)[0,1].
f (0)=ln2>0, f (1) = ln(1+e)–2
=ln(1+e) –lne2
ln(1+ex)=2x1.
f (x) f (x0) ( f (x) f (x0)),
f (x0)f (x)I().
f (x)[a, b], f (x)[a, b].
1. f (x)[a, b],
f (x)[a, b].

0
a
b
x
y
x1
x2
B
A
M1
M2
mc Mc,
2.
0
y
x
1
3.

.
0
y
x
1
–1
1

(1)
,
(1)n.
((1) x0 ), x0 (1)((1) x0 ).
, .
(1)I, (1)I.
(1) D(1).
[a, b]?
.
1.
, xn xn1[0, 1](n=2, 3, ). Sn(x) = xn ,
n ,

> 0, N > 0, n > N, |Sn(x0) – S(x0)|< .
, N , x0 , , x0 , N, N = N(, x0).
, x0 N, D.
1.
N = N(), n >N , xD
D S(x).
| Sn(x) Sm(x) | < , x D .
n > m ,
2.((Weierstrass))
(1) N > 0, n > N, | un(x) | an , x D . an ;
(2)
2.
:
2. .
x x0 ( x = x0 ). x0, an, n=0, 1, 2, ···.
, x0 = 0, x (x = 0).

, x = 0 a0 .
.
.
3.((Abel)),
(1) x = x0 ( x0 0), | x | < | x0 | x , , (| x0 | , | x0 | ) x .
(2) x = x0 , | x | > | x0 | x , , ( | x0 | , | x0 | ) x .
. x = x0 , (|x0| , |x0| ).
x = x0 , (, |x0|) ( | x0 |, +) .
x
x0
x0
0
x
x0
x0
0
: (1) x = x0 ( x0 0) , .
,

| x | > | x0 | x ,
x1 , | x1 | > | x0 | ,
.

x = r , .
x
0
r
r
x0
x0
(r, r)
, x = 0 , , r = 0.
x , r = +.

.
2.


, .
(2) = 0, | x | = 0 < 1, xR, .
r = +.
r = 0.
1. (x–x0)
r 0, |x–x0|<rx, (x0 – r, x0 + r).
x0 r
2.

.
.
x = 0 ,
x = 0 ,
6.
3.:
(1)
(2)
(3)
b0 0,
cn
a0 = b0c0
c0 , c1 , … , cn .