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wyklady.dvi
1 Definicja
y 0
f x 0
ǫ > 0
x ∈ (x 0 − δ, x 0 ) =⇒
δ > 0
f (x) ∈ (y 0 − ǫ, y 0 + ǫ).
lim
x!x 0
f (x) = y 0
x!x 0 −0 f (x) = y 0 .
lim
y 0 f x 0
ǫ > 0
δ > 0
x ∈ (x 0 , x 0 + δ) =⇒
f (x) ∈ (y 0 − ǫ, y 0 + ǫ).
lim
x!x 0
f (x) = y 0
x!x 0 +0 f (x) = y 0 .
lim
f
y 0
x 0
y 0
x!x 0 f (x) = y 0 .
+
+
y 0
f x 0
ǫ > 0
δ > 0
x
(x 0 − δ, x 0 ) ∪ (x 0 , x 0 + δ) f (x) ∈
(y 0 − ǫ, y 0 + ǫ)
x 0
+∞
f
x 0
M > 0
f (x) > M x ∈ (x 0 − δ, x 0 ).
lim
x!x 0
f (x) = +∞.
−∞
f
x 0
M > 0
δ > 0
f (x) <
−M x
∈ (x 0 − δ, x 0 ).
lim
x!x 0
f (x) = −∞.
lim
2 Definicja
δ > 0
+∞
f
x 0
M > 0
δ > 0
x
∈ (x 0 , x 0 + δ).
lim
x!x 0
f (x) = +∞.
−∞
f x 0
M > 0
δ > 0
f (x) <
−M x ∈ (x 0 , x 0 + δ).
lim
x!x 0
f (x) = −∞.
3 Definicja
f
+∞
x
→ +∞
M > 0
K > 0
f (x) > M
x > K.
lim
x!+1
= +∞.
f
−∞
x
→ +∞
M > 0
K > 0
f (x) <
−M
x > K.
lim
x!+1
= −∞.
f
+∞
x
→ −∞
M > 0
K > 0
f (x) > M
x <
−K.
lim
x!−1
= +∞.
f
−∞
x → −∞
M > 0
K > 0
f (x) <
−M
x <
−K.
lim
x!−1
= −∞.
f (x) > M
4 Definicja
y 0
+∞
−∞
x 0
±∞
x n < x 0
f x 0 ±∞
x n > x 0 lim
n!1
f (x) = y 0
+∞
(x n )
−∞
+
5 Przykład
f (x) = x 2
x = 1 1
1
3
ǫ
3
1, 1 + ǫ
3
ǫ <
δ =
x
∈
(1, 1 + δ) =
f (x) = x 2
> 1 2 = 1
1 + ǫ
3
2
= 1 + 2ǫ
3
(ǫ< 3 )
< 1 + 2ǫ
3
+ ǫ
f (x) <
+ ǫ 2
3 = 1 + ǫ,
f (x) ∈ (1, 1 + ǫ) ⊆ (1 − ǫ, 1 + ǫ)
+
ǫ
ǫ
≥ 1
9
1
10
δ =
f (x) = x 2
x = 1 1
ǫ < 1
2
δ = ǫ
2
1 − ǫ
x ∈ (1 − δ, 1) =
2 , 1
1 − ǫ
2
2
f (x) = x 2
>
= 1 − ǫ + ǫ 2
> 1 − ǫ
f (x) < 1 2 = 1,
f (x) ∈ (1 − ǫ, 1) ⊆ (1 − ǫ, 1 + ǫ)
f (x) = x 2
x = 1 1
f (x) =
1
x 2
x = 0
∞ M > 0
1
x 2
δ
> M 0 < x < δ
r
1
x 2
1
M
1
M ,
> M
⇐⇒
x 2
<
⇐⇒
x <
r
1
M
x ∈ (0, δ)
δ =
+
±∞
6 T WIERDZENIE
x!x 0 f (x) lim
lim
x!x 0 g(x)
47713883.019.png 47713883.020.png 47713883.021.png 47713883.022.png 47713883.001.png 47713883.002.png 47713883.003.png 47713883.004.png 47713883.005.png 47713883.006.png
x!x 0 (f (x) ± g(x)) = lim
x!x 0 f (x) ±
x!x 0 g(x)
lim
x!x 0 (f (x) g(x)) = lim
x!x 0 f (x)
x!x 0 g(x)
lim
x!x 0
f (x)
g(x)
= lim x!x 0 f (x)
lim x!x 0 g(x)
lim
x!x 0
g(x) = 0
x!x 0 f (x) g(x) =
x!x 0 f (x)
lim
lim x! x 0 f(x)
7 Definicja f
x 0
f (x 0 ) = lim
x!x 0 f (x)
8 Definicja
I
9 Przykład
x
∈
R
x →
x a
a
x ∈
R +
x
∈
R
x
∈
R +
x
∈
R
sinh cosh tgh ctgh
R +
R −
10 T WIERDZENIE f g
• f + g
g
− g
• f /g
lim
lim
lim
lim
• f
• f
47713883.007.png
+
11 Przykład
f (x) =
x − 1
x 2 + 2
x = 2
2 2 + 2 = 0
x!2 f (x) = f (2) =
lim
2 2 + 2 = 1
6 .
x = 2
f (x) = 3x 2 − 5x − 2
5x 2 − 20
x = 2
= 3(x − 2)(x + 3 )
5(x − 2)(x + 2) = 3(x + 3 )
x = 2
x − 2
5(x + 2)
x = 2.
→ 3(x + 3 )
5(x + 2)
x = 2
x
x!2 f (x) = lim
5(x + 2) = 3(2 + 3 )
5(2 + 2) =
7
20 .
x!2
1
x ,
1
x 2 ,
1
x 3
x = 0
c
0
2 − 1
3x 2 − 5x − 2
5x 2 − 20
3(x + 3 )
lim
±∞
47713883.008.png 47713883.009.png 47713883.010.png 47713883.011.png 47713883.012.png 47713883.013.png 47713883.014.png 47713883.015.png 47713883.016.png 47713883.017.png 47713883.018.png

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