Lösungen

1

Plot[Sqrt[1-Cos[x]^2],{x,0,5}];

Plot[Sin[x]/Sqrt[1-Cos[x]^2],{x,0,5}];

Plot[ArcCos[Sin[x]/Sqrt[1-Cos[x]^2]],{x,0,5},PlotRange->{-4,4}];

ArcCos[1]

ArcCos[-1]

Plot[ArcCos[Sin[x]/Sqrt[1-Cos[x]^2]],{x,0,5},PlotRange->{-4,4}];

Plot[ArcCos[Sin[x]/(Sqrt[1-Cos[x]^2]+1)],{x,0,5},PlotRange->{-4,4}];

2

(x+1)/x (x-1+(x^3-1)/(x-1)/(x+1))-2x//Simplify

3

Limit[(Cos[n]^3+3)/(Sin[n]+3n),n->Infinity]

Limit[1/n^2 (n-1)Log[n],n->Infinity]

Limit[(Cos[n]+n^3-2n^2+n-1-1/n)/(n Sin[n]-3n^3+n-5),n->Infinity]

Limit[(Cos[n]^2 Tan[1/n])/( Sin[n]+3)+(2n^2-3)/(3n^2+2),n->Infinity]

4

a Monoton (nicht streng), periodisch zu jeder Periode, gerade

Plot[Sign[x^2+1],{x,-3,3}];

b Monoton (nicht streng)

Plot[Sin[Sign[x]],{x,-3,3}];

c Nur stückweise monoton (streng), periodisch zu jeder Periode, gerade

Plot[Cos[Sin[x]],{x,-5,5}];

d Monoton (streng), ungerade

Plot[Log[10^x],{x,-3,3}];

e Monoton (streng)

Plot[Cosh[x]-Sinh[x],{x,-3,3}];

Plot[{Cosh[x]-Sinh[x], E^(-x)},{x,-3,3}];

f Nur stückweise monoton (streng), gerade

Plot[x Sin[x]+Abs[x] Cos[x],{x,-3,3}];

Plot[x Sin[x]+Abs[x] Cos[x],{x,-15,15}];