, x
sin (-x) = -sin x, cos (-x) = cos x
, y = sin x , y = cos x . x y = tg x tg (-x) = -tg x, y = tg x .
, x
sin (x + 2π) = sin x, cos (x + 2π) = cos x.
, 2π. 2π.
f (x) , T ≠ 0, x f (x - T) = f (x) = f (x + T).
T f (x).
, x f (x), x + T, x - T x + Tn, n Z, f (x + Tn) = f (x), n Z
2π y = cos x, y = sin x.
π - tg x.
R .
[-1; 1], .. .
: sin(−x)=−sin x ∈ R. .
2 π:
sin(x+2 π k) = sin x, k ∈ Z ∈ R.
sin x = 0 x = πk, k ∈ Z.
sin x > 0 () x ∈ (2πk, π+2πk), k ∈ Z.
sin x < 0 () x ∈ (π+2πk, 2π+2πk), k ∈ Z.
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R .
[-1; 1], .. .
: cos(−x)=cos x ∈ R. OY.
2 π:
cos(x+2 πk) = cos x, k ∈ Z ∈ R.
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, .. .
: tg(−x)=−tg x . π, .. tg(x+ πk) = tg x, k ∈ Z .
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, .. . : ctg(−x)=−ctg x . π, .. ctg(x+ πk)=ctg x, k ∈ Z .
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. .
. A f (x) a, a , , a, ε > 0 δ > 0 , x, | x a | < δ, x ≠ a, | f (x) A | < ε.
. A f (x) a, a , , a, , a, A.
1.3.6.1. y = x 2 x → 2. |
1.3.6.2. x → 0. |
A a, ,
.
1.3.6.3. y = { x (x ≠ 0); 1 (x = 0)} x → 0 0. |
a = 0 0: a = 0 0, ( ). a = 0 0, f (0) = 1.
f (x) a, .
A 1 f (x) a, ε > 0 δ > 0 ,
A 2 f (x) a, ε > 0 δ > 0 ,
a. . x → 0 : . ,
ε > 0 δ- a, x, | x a | < δ, x ≠ a, | f (x)| > ε, , f (x) a :
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|
, x = 0 , +∞ ∞. ,
ε > 0 δ > 0, x > δ | f (x) A | < ε, , f (x) x, , A:
x, : :
, , ε > 0 δ > 0, x > δ f (x) > ε. , ε > 0 δ > 0, x > δ f (x) < ε. , ε > 0 δ > 0, x < δ f (x) < ε.
f (x) a, a, f (, a ). , A ≠ 0, a, ( , a) f , A.
δ > 0, x, δ- a,
g (x) ≤ f (x) ≤ h (x), |
, |
δ > 0, x, δ- a,
f (x) < g (x), |
A ≤ B.