(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, 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.