4.1. = () → .
. , = () → 1 ≠ 2. (...) () = 1 + a1() () = 2 + a2(), a1(), a2() ... → . 1 + a1() = 2 + a2() 1 2 = a1() a2(). , . . , , .
4.2. = (), z = z (x) → , , , ,
1) ( () z (x)) = y (x) z (x);
2) (y (x) × z (x)) = y (x) × z (x),
, z (x) ≠ 0, ,
3) = .
. y (x) = , z (x) = .
... () = + a(), z (x) = B + b(x), a(), b() ... → . :
1) () z (x) = ( ) + (a() b()). ... a() b() ..., ( () z (x)) = , . . ( () z (x)) = y (x) z (x);
2) y (x) × z (x) = ( + B + b(x) = × + a() × B + × b(x) + a() × b(x). ... a() × B + × b(x) + a() × b(x) ... → .
(y (x) × z (x)) = = y (x) × z (x);
3) ≠ 0.
− .
... ... → .
= .
, = . , e, , , , , │ − │< , . . − < − < , < < . , . ...
... → .
a1(), . . = a1(). + a1(). ... = = .
4.1. , . . = const, ( × ()) = ().
4.2. () = , m ( ()) m = ( ()) m = Am.
4.3. u = u (x), v = v (x), y = y (x) , . u (x) £ y (x) £ v (x) u = u (x), v = v (x) → , = () → .
. u (x) = v (x) = A. . . u (x) £ y (x) £ v (x), u (x) − £ y (x) − £ v (x) − .
"e > 0 d1 > 0 d2 > 0 , 0 <│ − │< d1 │ u (x) − A │< e, 0 <│ − │< d2 │ v (x) − A │< e. d = min{d1, d2}. , 0 <│ − │< d −e < u (x) − A < e −e < v (x) − A < e. u (x) − £ y (x) − £ v (x) − −e < (x) − A < e, . . │ (x) − A │< e. , () = .
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4.4. = f (x) , . → = f (x) () , , ().
. f (x) = . , "e > 0 d > 0, , 0 <│ − │< d, │ f (x) − A │< e, . . −e < f (x) − A < e.
> 0, e = A − e < f (x), f (x) > A − e = = A − A = A > 0, . . f (x) > 0 −d < x − < d, . . − d < x < d + .
< 0, e = − , f (x) < A + e f (x) < A + e = = A − A = A < 0, . . f (x) < 0 − d < x < d + .
, .
4.5. u (x), v (x) , , , = , u (x) < v (x), u (x) v (x) → . u (x) £ v (x).
. u (x) = , v (x) = . , > B. 4.2 (u (x) − v (x)) = > 0. 4.4 , u (x) − v (x) > 0, . . u (x) > v (x), .
, £ , . . u (x) £ v (x).
4.1. , f (x)/ g (x) ( ) → , ( ) → . f (x)/ g (x) → : , , , , . f (x)/ g (x), , , . .