2.
12.
12.1..
12.2. .
12.3. .
12.4.
, , .
. . . .
.. 6 . 122-160
.., .. VII .95-133
.. , .. , .. . II I 1.1 - .330-336 1.2. 2.1. ε δ (. 148-155)
.. 8 II .33-59
., . ? VI, , 1 (.300-317), 2 (.317-330), 3 (.330-337); VIII 4 . 461-463
1. ? ?
2. () () ? ? ?
x 0. .
3.
4.
5.
6. ?
7. ?
8.
12.1.
e . e- x 0 x, (x 0 ‑ e, x 0 + e), x 0. x e ‑
0 < ê x x 0ç < e.
(x 0 ‑ e, x 0 + e) \ { x 0} x 0 ( x 0 )
∞, : (ε>0)
x 0 = +∞ Ue ( +∞ )= (ε, +∞]
x 0 = - ∞ Ue ( - ∞ ) = [- ∞, ε)
: Ue (a) - e - , - .
y ε-, δ- (U δ (a))
|
|
.
y = x 2 x 0 = 2. 4.
.
- e e - y 0 = 4. , x 0 = 2 ( 1 d), x , y, x 2, e - y 0 = 4. , e. x 0 = 2 . .
. x 0 = 2. x 0 ¹ 2 :
.
2. x 0 = 2 3 , y 0 = 3 . e, , x, x 0 = 2 ( x 0 = 2, e), y e- y 0 = 3. , e.
12.2(1).
. A y = f (x) x 0 ( , x, x 0, → x 0) e d, x d - x 0 y e - y = A.
-. A y = f (x) x 0, e d, x,
0 < ê x x 0ê < d,
ê y A ê < e.
, A y = f (x) x = x 0, f(x) →A x → x 0
12.2.
.
, , x = x 0, , ( x 0 )
. , x > 0, y = 2 x; x < 0, y = 2 x; x = 0 .
3. , , , x = 0 .
(..4)
.4.
12.2(1).
,
ε > 0. ε δ > 0, x ≠ x 0, | x x 0| < δ, .. 0 < | 2| < δ |f(x) A|<ε, .. |(2 + 1) 5 | < ε.
|
|
|2( 2)| < ε, ..
| 2| < ε/2. , δ = ε/2 | 2| < δ |f() 5| < ε ( , , , ). , .
.
1. .
2. , C .
3. C ,
.
4. , , , , . , , .
12.2(2)
1.
2. ( . 12.4): , .
, .. =2. → 2, :
, ,
3. ( - )
4.. , , .. .
( , .. ) :
12.2(2)
B f (x) a ( ), e d, 0 < x a < d ê B f (x) ê < e.
. , x = 0 .
f (x) b ( ), e d , 0 < b x < d ê C f (x)ê < e.
, ( , 3) x = 0:
; .
12.2 ( ) .
, , , :
;
. f (x), (d; ¥). f (x) , :
,
e δ, , δ, :
½ f (x) A ½ < e.
f (x)
(¥; d). f (x) , :
,
e δ, , , δ, :
½ f (x) A ½ < e.
+∞ (.. f (x) →+ ∞ x → x 0)
+∞ f (x) x → x 0, e d, x, 0 < ê x x 0ê < d,
f(x) > e.
= - ∞ (.. f (x) →- ∞ x → x 0)
+∞ f (x) x → x 0, e d, x, 0 < ê x x 0ê < d,
|
|
f(x) < - e.
.
1. . , .
2. . e , 2,72.
, :
12.2
1.
2.
12.2.
:
12.3.
:
, .
12.3.
1.
, ( 1- )
2.
,
12.3(2)
f , |f| →+∞. f→+∞, f , f→ - ∞, f . , f , , , f(x) = x (-1)[x], ,
|f| →+∞.
, f x 0 , 1/f - x 0.
, , - .
, , ( ), / , / , .
, f →+∞, g→- ∞, : , , . .
() .
: - ∞.
12.4.
1.∞-∞, x→a
=+∞
f(x)=x+10 →+∞
g(x)= -x → - ∞
f+g = 10 → 10
2..∞-∞, x→a
=+∞
f(x)=x+sin x→+∞
g(x)=-x → - ∞
f+g=sin x,
3.
=+∞
f(x) = 1/x →0
g(x) = x→+∞
fg = 1→1
a
f(x) = 20/x →0
g(x) = x→+∞
fg = 20→20