Lösungen / Solutions

1

a

f[x_]:=E^(-x^2+4x+1)

Limit[f[x],x-> Infinity]

Limit[f[x],x-> -Infinity]

b

D[f[x],x]

solv1=Solve[Evaluate[D[f[x],x]]==0,{x}]//Flatten

x1=x/.solv1

c

solv0=Solve[f[x]==0,{x}]//Flatten

d

D[f[x],{x,2}]

solv2=Solve[Evaluate[D[f[x],{x,2}]]==0,{x}]//Flatten

x2=x/.solv2[[1]]

N[%]

x3=x/.solv2[[2]]

N[%]

e

Options[Plot]

Plot[f[x],{x,-1,4},Epilog->{PointSize[0.03],Point[{x1,f[x1]}],Point[{x2,f[x2]}],Point[{x3,f[x3]}]}];

2

a

g[x_]:=(2x-3)

f[x_]:=g[x]+1/x-2/x^2

Limit[f[x],x-> Infinity]

Limit[f[x],x-> -Infinity]

b

Limit[f[x]-g[x],x-> Infinity]

Limit[f[x]-g[x],x-> -Infinity]

c

D[f[x],x]

solv1=Solve[Evaluate[D[f[x],x]]==0,{x}]//Flatten

x1=x/.solv1[[1]]

N[%]

d

solv0=Solve[f[x]==0,{x}]//Flatten

N[%]

x0=x/.solv0[[1]]

d

D[f[x],{x,2}]

solv2=Solve[Evaluate[D[f[x],{x,2}]]==0,{x}]//Flatten

x2=x/.solv2

e

Plot[{f[x],g[x]},{x,-5,8},
Epilog->{PointSize[0.03],Point[{x0,f[x0]}],Point[{x1,f[x1]}],Point[{x2,f[x2]}]},
PlotRange->{-20,20}];

Plot[{f[x],g[x]},{x,1.5,10},
Epilog->{PointSize[0.03],Point[{x0,f[x0]}],Point[{x1,f[x1]}],Point[{x2,f[x2]}]}];

Plot[{f[x],g[x]},{x,4,5},
Epilog->{PointSize[0.03],Point[{x0,f[x0]}],Point[{x1,f[x1]}],Point[{x2,f[x2]}]}];

f[x] - g[x]  > 0   ??

Plot[Sign[f[x]-g[x]],{x,0,8}];

3

a

Remove[f]

f[x_]:=(4 E^x - 4)/(3 Sin[x]-4 Sin[2x]);
Limit[f[x],x-> 0]

b

Remove[f]

?Limit

f[x_]:=x^x;
Limit[f[x],x-> 0,Direction->-1]

c

Remove[f]

f[x_]:=(7 Log[x])/(8 Cot[x]);
Limit[f[x],x-> 0,Direction->-1]

d

Remove[f]

Table[Limit[Log[x]/x^n,x-> 0,Direction->-1], {n,0.1,2,0.1}]

Achtung! Attention!

IntegerQ[3] && Positive[3]

f[x_,n_ ]:=Log[x]/x^n /; IntegerQ[n] && Positive[n] ; f[x,3]

f[x_,n_ ]:=Log[x]/x^n /; IntegerQ[n] && Positive[n] ;  Plot[f[x,5],{x,1,2}];

Limit[Evaluate[f[x,2]],x-> 0,Direction->-1]

lim[n_]:=Limit[Evaluate[f[x,n]],x-> 0,Direction->-1];
lim[2]

lim[0.9]

Do[f[x_,n_]:=Log[x]/x^n /; IntegerQ[n] && Positive[n] ;lim[n_]:=Limit[Evaluate[f[x,n]],x-> 0,Direction->-1];Print[lim[n]],{n,1,3,0.1}];

Table[lim[n], {n,0.1,2,0.1}]

Table[lim[Evaluate[n]], {n,0.1,2,0.1}]

e

Remove[f]

Table[Limit[Log[x]/x^n,x-> Infinity], {n,0.1,2,0.1}]

f

Remove[f]

Table[Limit[x^n / E^x ,x-> Infinity], {n,0.1,2,0.1}]

g

Achtung! Attention!   x^n ? ??? x < 0

Remove[f]

Table[Limit[x^n / E^x ,x-> -Infinity], {n,0.1,2,0.1}] // Chop