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Sinuswundern x^k Sin(1/x)   
Prof. Dr. Dörte Haftendorn Ha 6/99 6/01 7/03

f[x_] := Sin[1/x] ; f[0] = 0 ; f '[x] f[h]/h (* Differenzenquotient für x = 0 *) f ''[x] ... 1}, {x, -3, 3}, PlotStyle -> farbig, PlotRange -> {-1, 1.5}, AspectRatio -> Automatic] ;

-cos(1/x)/x^2

sin(1/h)/h

(2 cos(1/x))/x^3 - sin(1/x)/x^4

[Graphics:index_6.gif]

Limit[f[x], x -> 0] (* nicht stetig in x = 0 *) <br /> Limit[f '[x], x -> 0] <br /> Limit[f[h]/h, h -> 0] (* Differenzialquotient für x = 0 ex . nicht *)

Interval[{-1, 1}]

Interval[{-∞, ∞}]

Interval[{-∞, ∞}]

h[x_] := x Sin[1/x] h '[x] h[k]/k (* Differenzenquotient für x = 0 *) h ''[x] hGraph = Pl ... x}, {x, -3, 3}, PlotStyle -> farbig, PlotRange -> {-1, 1.5}, AspectRatio -> Automatic] ;

sin(1/x) - cos(1/x)/x

sin(1/k)

x ((2 cos(1/x))/x^3 - sin(1/x)/x^4) - (2 cos(1/x))/x^2

[Graphics:index_15.gif]

Limit[h[x], x -> 0] Limit[h '[x], x -> 0] Limit[h[k]/k, k -> 0] (* Differenzialquotient für x = 0 ex . nicht *) Limit[h ' '[x], x -> 0]

0

Interval[{-∞, ∞}]

Interval[{-1, 1}]

Interval[{-∞, ∞}]

•Asyptotengewinnung aus der Taylorreihe

•Tranformation xSin[1/x] - asy(x) Overscript[->, x -> ∞] 0 in 1/z Sin[z] - asy[1/z] Overscript[->, z -> 0] 0

tay[z_] := Normal[Series[Sin[z], {z, 0, 6}]]

Clear[as] ; (1/z tay[z] // Expand) - as

z^4/120 - z^2/6 - as + 1

•Wenn dieser Term für z->0 gegen 0 gehen soll, muss  gelten:

as = 1 /. z -> 1/x

1

hAsyGraph = Plot[{h[x], x, -x, 1}, {x, -3, 2}, PlotStyle -> farbig, PlotRange -> {-1, 2}, AspectRatio -> Automatic] ;

[Graphics:index_29.gif]

Limit[h[x] - 1, x -> ∞] (* Asymptote y = 1 *) Limit[h '[x], x -> ∞] Limit[h ''[x], x -> ∞]

0

0

0

hh[x_] := x^2 Sin[1/x] hh '[x] hh[h]/h (* Differenzenquotient für x = 0 *) hh ''[x]

2 x sin(1/x) - cos(1/x)

h sin(1/h)

((2 cos(1/x))/x^3 - sin(1/x)/x^4) x^2 + 2 sin(1/x) - (4 cos(1/x))/x

hhGraph = Plot[{hh[x], x^2, -x^2}, {x, -2, 2}, PlotStyle -> farbig, PlotRange -> {-1, 1}, AspectRatio -> Automatic] ;

[Graphics:index_39.gif]

Limit[hh[x], x -> 0] Limit[hh '[x], x -> 0] <br /> (* nicht stetig in x = 0 *) Limit[hh[h]/h, h -> 0] (* Differenzialquotient für x = 0 ex . *) Limit[hh ' '[x], x -> 0]

0

Interval[{-1, 1}]

0

Interval[{-∞, ∞}]

•Asyptotengewinnung aus der Taylorreihe

•Tranformation x^2 Sin[1/x] - asy(x) Overscript[->, x -> ∞] 0 in 1/z^2 Sin[z] - asy[1/z] Overscript[->, z -> 0] 0

tay[z_] := Normal[Series[Sin[z], {z, 0, 7}]]

Clear[ass] ; (1/z^2 tay[z] // Expand) - ass

-z^5/5040 + z^3/120 - z/6 - ass + 1/z

•Wenn dieser Term für z->0 gegen 0 gehen soll, muss  gelten:

ass = (1/z) /. z -> 1/x

x

hhAsyGraph = Plot[{hh[x], x^2, -x^2, x}, {x, -3, 3}, PlotStyle -> farbig, PlotRange -> {-3, 3}, AspectRatio -> Automatic] ;

[Graphics:index_53.gif]

Limit[hh[x] - x, x -> ∞] (* Asymptote y = x *) Limit[hh '[x] - 1, x -> ∞] Limit[hh ''[x], x -> ∞]

0

0

0

hhh[x_] := x^3 Sin[1/x] hhh '[x] hhh[h]/h (* Differenzenquotient für x = 0 *) hhh ''[x]

3 x^2 sin(1/x) - x cos(1/x)

h^2 sin(1/h)

((2 cos(1/x))/x^3 - sin(1/x)/x^4) x^3 + 6 sin(1/x) x - 6 cos(1/x)

hhhGraph = Plot[{hhh[x], x^3, -x^3}, {x, -.1, .1}, PlotStyle -> farbig (* , PlotRange -> {-0.1, 1}, AspectRatio -> Automatic *)] ;

[Graphics:index_63.gif]

Limit[hhh[x], x -> 0] <br /> Limit[hhh '[x], x -> 0] (* stetig in x = 0 *) <br /> Limit[ ... ferenzialquotient für x = 0 *) Limit[hhh ' '[x], x -> 0] (* Unbeschränkt in x = 0 *)

0

0

0

Interval[{-∞, ∞}]

•Asyptotengewinnung aus der Taylorreihe

•Tranformation x^3 Sin[1/x] - asy(x) Overscript[->, x -> ∞] 0 in 1/z^3 Sin[z] - asy[1/z] Overscript[->, z -> 0] 0

tay[z_] := Normal[Series[Sin[z], {z, 0, 7}]]

Clear[asss] ; (1/z^3 tay[z] // Expand) - asss

-z^4/5040 + z^2/120 - asss - 1/6 + 1/z^2

•Wenn dieser Term für z->0 gegen 0 gehen soll, muss  gelten:

asss = (1/z^2 - 1/6) /. z -> 1/x

x^2 - 1/6

hhhAsyGraph = Plot[{hhh[x], x^3, -x^3, x^2 - 1/6}, {x, -1, 1}, PlotStyle -> farbig, PlotRange -> {-0.1, 1}, AspectRatio -> Automatic] ;

[Graphics:index_77.gif]

Limit[hhh[x] - x^2 + 1/6, x -> ∞] (* Asymptote y = x^2 - 1/6 *) Limit[hhh '[x] - 2 x, x -> ∞] Limit[hhh ''[x] - 2, x -> ∞]

0

0

0

hhhh[x_] := x^4 Sin[1/x] hhhh '[x] hhhh[h]/h (* Differenzenquotient für x = 0 *) hhhh ''[x]

4 x^3 sin(1/x) - x^2 cos(1/x)

h^3 sin(1/h)

((2 cos(1/x))/x^3 - sin(1/x)/x^4) x^4 + 12 sin(1/x) x^2 - 8 cos(1/x) x

hhhhGraph = Plot[{hhhh[x], x^4, -x^4}, {x, -.1, .1}, PlotStyle -> farbig (* , PlotRange -> {-0.1, 1}, AspectRatio -> Automatic *)] ;

[Graphics:index_87.gif]

Limit[hhhh[x], x -> 0] Limit[hhhh '[x], x -> 0] Limit[hhhh[h]/h, h -> 0] (* Differenzialquotient für x = 0 *) Limit[hhhh ' '[x], x -> 0]

0

0

0

Indeterminate

•Asyptotengewinnung aus der Taylorreihe

•Tranformation x^4 Sin[1/x] - asy(x) Overscript[->, x -> ∞] 0 in 1/z^4 Sin[z] - asy[1/z] Overscript[->, z -> 0] 0

tay[z_] := Normal[Series[Sin[z], {z, 0, 7}]]

Clear[assss] ; (1/z^4 tay[z] // Expand) - assss

-z^3/5040 + z/120 - assss - 1/(6 z) + 1/z^3

•Wenn dieser Term für z->0 gegen 0 gehen soll, muss  gelten:

asss = (1/z^3 - 1/(6 z)) /. z -> 1/x

x^3 - x/6

hhhhAsyGraph = Plot[{hhhh[x], x^4, -x^4, x^3 - 1/6 x}, {x, -0.7, 0.7}, PlotStyle -> farbig, PlotRange -> {-0.3, .3}] ;

[Graphics:index_101.gif]

Limit[hhhh[x] - x^3 + 1/6 x, x -> ∞] (* Asymptote y = x^3 - 1/6 x *) Limit[hhhh '[x] - 3 x^2 + 1/6, x -> ∞] Limit[hhhh ''[x] - 6 x, x -> ∞]

0

0

0

hhhhh[x_] := x^5 Sin[1/x] hhhhh '[x] hhhhh[h]/h (* Differenzenquotient für x = 0 *) hhhhh ''[x]

5 x^4 sin(1/x) - x^3 cos(1/x)

h^4 sin(1/h)

((2 cos(1/x))/x^3 - sin(1/x)/x^4) x^5 + 20 sin(1/x) x^3 - 10 cos(1/x) x^2

hhhhhGraph = Plot[{hhhhh[x], x^5, -x^5}, {x, -.1, .1}, PlotStyle -> farbig (* , PlotRange -> {-0.1, 1}, AspectRatio -> Automatic *)] ;

[Graphics:index_111.gif]

Limit[hhhhh[x], x -> 0] Limit[hhhhh '[x], x -> 0] Limit[hhhhh[h]/h, h -> 0] (* Differenzialquotient für x = 0 *) Limit[hhhhh ' '[x], x -> 0]

0

0

0

0

•Asyptotengewinnung aus der Taylorreihe

•Tranformation x^5 Sin[1/x] - asy(x) Overscript[->, x -> ∞] 0 in 1/z^5 Sin[z] - asy[1/z] Overscript[->, z -> 0] 0

tay[z_] := Normal[Series[Sin[z], {z, 0, 7}]]

Clear[asssss] ; (1/z^5 tay[z] // Expand) - asssss

-z^2/5040 - asssss + 1/120 - 1/(6 z^2) + 1/z^4

•Wenn dieser Term für z->0 gegen 0 gehen soll, muss  gelten:

asssss = (1/z^4 - 1/(6 z^2) + 1/120) /. z -> 1/x

x^4 - x^2/6 + 1/120

hhhhhAsyGraph = Plot[{hhhhh[x], x^5, -x^5, x^4 - 1/6 x^2 + 1/120}, {x, -0.5, 0.5}, PlotStyle -> farbig, PlotRange -> {-0.001, .02}] ;

[Graphics:index_125.gif]

Limit[hhhhh[x] - x^4 + 1/6 x^2 - 1/120, x -> ∞] (* Asymptote y = x^4 - 1/6 x^2 + 1/12 ... t[hhhhh '[x] - 4 x^3 + 1/3 x, x -> ∞] Limit[hhhhh ''[x] - 12 x^2 + 1/3, x -> ∞]

0

0

0

Converted by Mathematica  (July 27, 2003)