(.)
-
()
.
..
2011
1.
.1.
n an, , { an }:
a1, a2,,an.
, an=f(n).
a1, a2,,an , an n - .
1: ) 2, 4, 6, 2 n, ( );
) 1, 0, 1, 0, ( );
) 0, , , , , , , - , . . n 1. , , , , , .. n , .
{ an }, e > 0 N(e), e, n > N(e) | an - A | < e.
. (): " (), $ (), Û ( ), :
(A = an)Û ("e > 0 $ N(e): " n > N(e) | an A | < e)
, , , .
: n { an } .
.2.
y = f(x) , ( ), e > 0 S > 0, | | > S | f(x) A | < e.
f(x) = A, f(x) A ¥ ().
(A = f(x) Û ("e > 0 $ S = S(e) > 0: " x: | x | > S | f(x) A | < e).
: f(x) .
f(x) 0, ( 0),
"e > 0 $ d = d(e) > 0: " x ¹ x 0, | x0 | < d | f(x) A | < e.
: , 0, f(x) .
1. 0, , 0, 0. 0 0, 0.
|
|
2. 0 , 0, f(x) , : f(x) = A.
, 0 > 0, , .: f(x) = A
, f(x) = f(x) = , f(x) = A.
. 3.
a (x) 0 ( ¥), : a (x) = 0.
, - ... ¥, - ... 0.
:
: f(x) = A, f(x)
f(x) = A + a(x), a(x) 0 (¥).
: f(x) a (x) 0 (¥), 0 (¥), ..
f(x) = A.
:
1. .
2. .
3. , , .
. 3 a (x) b (x) - . , ¹ 0 ¥. =1 a (x) b (x) a (x) b (x).