y=f(x) x→a x →∞, , .. , .
:
1) , .
. . f(x)=α(x)+β(x), . , ε > 0 δ> 0, , x, |x a|<δ, |f(x)|< ε.
, ε > 0. α(x) , δ1 > 0, |x a|< δ1 |α(x)|< ε / 2. , β(x) , δ2 > 0, |x a|< δ2 | β(x)|< ε / 2.
δ=min{ δ1, δ2 }. a δ |α(x)|< ε / 2 | β(x)|< ε / 2. ,
|f(x)|=| α(x)+β(x) | ≤ |α(x)| + | β(x)| < ε /2 + ε /2= ε,
.. |f(x)|< ε, .
2) a(x) f(x) x→a ( x→∞) .
. f(x) , , x a|f(x)|≤M. , a(x) x→a, ε>0 a, |α(x)|< ε/M. | αf|< ε/M= ε. , af . x→∞ .
3) α(x) f(x), , .
.
. 1 /f(x) . , .. .
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( )
1. , , ..
.
. , . . f(x)=b+α(x) g(x)=c+β(x), α β . ,
f(x) + g(x)=(b + c) + (α(x) + β(x)).
b + c , α(x) + β(x) ,
.
|
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2. , :
.
. . , f(x)=b+α(x) g(x)=c+β(x)
fg = (b + α)(c + β) = bc + (bβ + cα + αβ).
bc . bβ + c α + αβ . .
1. :
.
2. :
.
3. , , ..
.
. . , f(x)=b+α(x) g(x)=c+β(x), α, β .
.
, , c2≠0.
4. f(x), u(x) v(x), u (x)≤f(x)≤ v(x). u(x) v(x) x→a ( x→∞), f(x) , ..
, .
5. x→a ( x→∞) y=f(x) y≥0 b, : b≥0.
. . , b<0, |y b|≥|b| , , x→a. y b x→a, .
6. f(x) g(x) x f(x)≥ g(x) , b≥c.
. f(x)-g(x) ≥0, , 5 , .
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