8 Grenzwerte f
Embed Size (px)
Citation preview
-
7/24/2019 8 Grenzwerte f
1/12
y = f(x)
f(x)
x0
f(x) =x2
x0= 1
(an) n
x0
x0
x0
http://0.0.0.0/http://0.0.0.0/
-
7/24/2019 8 Grenzwerte f
2/12
x0
MR f :MR z
x0 (xn)nN M f
xn=x0
n
Aus limn
xn= x0 folgt limn
f(xn) =z .
limxx0
f(x) =z .
x0 R z +
x x0 f(x) z f(x0) =z
xn n x0
f(x) =x
f(x) = x xk (xn)nN xk =f(xk)
f
x0 R
limxx0
f(x) = limxx0
x= x0
f x0
f(x0) x0 x0
x0= 0
f f(x) = 0 x = 0 f(0) = 1
limx0f(x) = 0 f(0) = 1
x0= 0
f : R+ R, x x1 limx0f(x) =
(xn) xn R+ limn xn = 0 limn f(xn) = x= 0
x0 = 0
-
7/24/2019 8 Grenzwerte f
3/12
y
1
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
y
x
-1
1
sign(x) R
sign(x) =
+1 fur x >0
0 fur x= 01 fur x x0
limxx+
0
f(x)
x = 0
1 1 f x0 = 0
-
7/24/2019 8 Grenzwerte f
4/12
f
x0
f
x0
xn
n x0 = 0
x0
f
< x0<
limxx0
|f(x)|= + .
x0 R
Konstanter Faktor: limxx0
af(x) =a limxx0
f(x) fur allea R
Summe: limxx0
[f(x)+ g(x)] = limxx0
f(x) + limxx0
g(x)
Produkt: limxx0
[f(x) g(x)] = limxx0
f(x) limxx0
g(x)
Quotient: limxx0
f(x)
h(x) =
limxx0f(x)
limxx0h(x)
falls limxx0
h(x)= 0
Betrag: limxx0
(|f(x)|) =| limxx0
(f(x))|
http://0.0.0.0/
-
7/24/2019 8 Grenzwerte f
5/12
f(x) = 5x x0 R
limxx0 f(x) = limxx0 5x= 5 lim
xx0x
= 5x0
f(x) =x2. x0 R
limxx0
f(x) = limxx0
x2
= limxx0
x limxx0
x
= x
2
0
f(x) = xn n 0 x0 = 0 f(x) =
n=0ax
N a R limxx0f(x) limxx0f(x) =
f(x0)
x0
f x z
(xn) limn xn=
limn
f(xn) =z
limx
f(x) =z .
f(x) x
limx
f(x)
limx
f(x)
a= 0 b= 0 c= 0
limx
ax
bx+c
= limx
a
b+c x1
= a
b+c limx x1
= a
b+c 0
= a
b
-
7/24/2019 8 Grenzwerte f
6/12
f
M R f : M R 0
x0 M f(x0)
limxx0
f(x) =f(x0).
f
x0
x0 M
f
R
f, g: MR x0 a R
Konstanter Faktor: af ist an der Stelle x0 stetig.
Summe: f+ g ist an der Stelle x0 stetig.
Produkt: f g ist an der Stelle x0 stetig.
Quotient: f
g ist an der Stelle x0 stetig, falls g(x0)= 0.
Komposition : g f ist an der Stelle x0 stetig,
falls f(x) an der Stelle x = x0 stetig ist
und g (z)an der Stellez= f(x0) stetig ist.
gf f g
x g(f(x))
limxx0
(g f)(x) = limxx0
g(f(x))
= g( limxx0
f(x))
= g(f( limxx0
x))
= g(f(x0))
= (g f)(x0)
g f
http://0.0.0.0/
-
7/24/2019 8 Grenzwerte f
7/12
f :R R, xxk k N
f(x) = x
f(x) =x2
f(x) =x3
k f(x) = xk
f(x) = xk k N
f(x) =axk a R
f :R R, xn
k=0akxk
ak R
f :R R, x |x| x0= 0
x x x
x0 f f(x0) = 0
a, b R a < b f : [a, b] R f(a) < 0 f(b) > 0
f(a)> 0
f(b)< 0
f
[a, b]
x0
x01x02
x03
y
xa b
f(a)
f(b)
0
-
7/24/2019 8 Grenzwerte f
8/12
f
f
a, b R a < b f : [a, b] R f f(a)
f(b)
d
f(a)
f(b)
c
a < c < b
f(c) =d
yf(b)
a
f(a)
x
d
bc
g(x) = f(x) d g(a) > 0 g(b) < 0 g(a) < 0 g(b)> 0
g
f(a)
f(b)
f : M R
max f
maxxMf(x) max{f(x) :x M}
f :MR
min f minxMf(x) min{f(x) :x M}
f : R+ R, f(x) = x
x= 0
f
-
7/24/2019 8 Grenzwerte f
9/12
1
2
3
4
1 2 3 4
1
2
3
4
1 2 3 4
1
2
3
4
1 2 3 4
1
2
3
4
1 2 3 4
f(x) = x1
f : R+ R, x x1
(0, 1)
f : (0, 1) R, x x1
(0, 1]
f : (0, 1] R, x x1
[1/2, 1]
2
f : [1/2, 1] R, x x1
-
7/24/2019 8 Grenzwerte f
10/12
a, b R a < b f : [a, b] R
x1 [a, b] x2 [a, b]
y
a xb
Max
Min
[a, b]
a b f :R+ R, f(x) =x
(an)nN (bn)nN a1 = a b1 = b
an+1 bn+1 an bn
an+bn2
x an bn
fan+bn
2
> 0
an+1= an bn+1 = an+bn
2 .
fan+bn
2
= 0
fan+bn
2
< 0
an+1= an+bn
2 bn+1 = bn
(an)nN (bn)nN
p
limn
an = limn
bn= p
p
f(an) 0 n N 0 limn f(an) f limn f(an) =f(limn an) = f(p) 0
f(p)
(bn)nN
0 f(p)
f(p) = 0 p
http://0.0.0.0/
-
7/24/2019 8 Grenzwerte f
11/12
f(x) =
2 ; x= 31 ; x= 3
f(x) x= 3
f(x)
f(x)
x 3
g(x) = 1
x 2+ 1
h(x) = 1/x
g(x)
limx2
g(x)
limx2+
g(x)
limx2
g(x) limx1
g(x)
sinh(x)
cosh(x)
sinh(x) =ex ex
2 cosh(x) =
ex + ex
2
sinh(x)
cosh(x)
x= 0
x
limx0
2x2
limx1
1
(x 1)3
limx
2x4 + 5x3
x7
limx
8x21 + 2x13 20x3
4x21 x4
limx0
8x21 + 2x13 20x3
4x21 x4
limxx0
x2 x20x x0
-
7/24/2019 8 Grenzwerte f
12/12
