1. , - a, .
. a , , B > 0,
(4) |
x,
(5) |
, ε > 0 ,
(6) |
x,
(7) |
δ.
(8) |
, (5) (7) (4) (6).
, ε > 0
x δ- a.
2. .
. ε > 0 ; . ,
(9) |
(10) |
, (9) (10) , ,
. .
6.) : , .
1 ( e-d).
f(x) x->x0 ( x0 ) ">0, $ d , U*(x0, d), |f(x)|>M.
: |f(x)|>M , f(x)U(∞;1/M)=> : =∞; f(x)-> ∞, x->x0
2( )
f(x) .. x->x0, {xn} , 0, {f(xn)} ∞
1 2
:
1) f(x) .. x->x0 , x->x0.
f(x) .. x->x0 , x->x0 ( .)
2) f(x) g(x) .. , f(x)+g(x) .. x->x0.
3) f(x) .. x->x0, g(x) U*(x0 , d), f(x)+g(x) .. x->x0.
4) f(x) g(x) .. , f(x)*g(x) .. x->x0.
5) f(x) . x->x0, g(x) x->x0 , =a≠0, f(x)*g(x)- . x->x0
6) f(x) . x->x0, "xU*(x0 , d) |f(x)|≤|g(x)|, g(x)- .. ->0
|
|
7) f(x) g(x)-.. x->x0 $ d , f(x)≤£(x)≤g(x), "xU*(x0 , d), £(x)- .. x->x0 ( 2 )
7.) . .
AÎℝ f(x) x, x0 ( x0 ), "e>0 $d>0 , x
0 < x 0 x < d, f (x)ÎU(A, e).
2) B Îℝ f (x) x, x 0 , "e>0 $d>0 , x
0 < x x 0 < d, f (x)ÎU(B, e).
3) , f (x) x 0 +¥ ( ¥ ) ( +¥ ( ¥ ) x, - x 0 ), " M >0 $d>0 , x 0 < x 0 x < d, f (x) > M (f (x) < M).
4) , f (x) x 0 +¥ ( ¥ ), " M >0 $d>0 , , x 0 < x x 0 < d, f (x) > M (f (x) < M).
5 ( f (x) x x 0 x 0Îℝ).
f (x) () x x 0Û f (x) x x 0.
1)
lim f(x)=A (x->x0), , f(x)
={x , f(x) => lim f(x)=A (x-> ) lim f(x)=A (x-> )
2
lim f(x)(x-> ) = lim f(x) (x-> ) =A
f(x) , f(x)
lim f(x)=A (x-> ) f(x)
8.)