{NotebookFileName[],DateString[]}

style[
t_
]:=
Style[t,Bold,FontFamily->"Arial",FontSize->14]

Entropy

s1[
p_
]:=
-p Log[p]-(1-p)Log[1-p]

s1[0]:=0

s1[1]:=0

Plot[s1[p],{p,0,1}]

D[(1/(Log[p])),p]

(0/0) forms

Block[{nd=4,l=0,v=x,n=(E^x)-1,d=x,ds},
ds=Table[
{D[n,{v,i}],D[d,{v,i}]}->({D[n,{v,i}],D[d,{v,i}]}/.v->l),
{i,0,nd}
];
AppendTo[ds,Limit[(n/d),v->l]];
ds
]//Column

Block[{nd=4,l=0,v=x,n=ArcSin[x],d=ArcTan[x],ds},
ds=Table[
{D[n,{v,i}],D[d,{v,i}]}->({D[n,{v,i}],D[d,{v,i}]}/.v->l),
{i,0,nd}
];
AppendTo[ds,Limit[(n/d),v->l]];
ds
]//Column

Series[ArcCos[x],{x,1,3}]

Series[(((x-1))^(1/2)),{x,1,3}]

Limit[((ArcCos[x])/(Sqrt[1-(x^2)])),x->1, Direction->-1]

Block[{nd=4,l=1,v=x,n=ArcCos[x],d=(Sqrt[1-(x^2)]),ds},
ds=Table[
{D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}->({D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}/.v->l),
{i,0,nd}
];
AppendTo[ds,Limit[(n/d),v->l]];
ds
]//Column

Repeated use

Block[{nd=4,l=0,v=x,n=x (E^x)-(E^x)+1,d=(x^2),ds},
ds=Table[
{D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}->({D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}/.v->l),
{i,0,nd}
];
AppendTo[ds,Limit[(n/d),v->l]];
ds
]//Column

Block[{nd=4,l=0,v=x,n=1-Cos[x],d=(x^2),ds},
ds=Table[
{D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}->({D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}/.v->l),
{i,0,nd}
];
AppendTo[ds,Limit[(n/d),v->l]];
ds
]//Column

Series[Cos[x],{x,0,7}]

SeriesCoefficient[Cos[x],{x,0,n}]

Series[1-Cos[x],{x,0,7}]

The e limit: (1^∞) indeterminate form

Limit[(((1+x/n))^n),n->∞]

f(x,n)=(((1+(x/n)))^n)
log(f(x,n))=n log(1+(x/n))=((log(1+x/n))/(1/n))

D[Log[1+(x/n)],n]

D[1/n,n]

((D[Log[1+(x/n)],n])/(D[1/n,n]))

((D[Log[1+(x/n)],n])/(D[1/n,n]))//Simplify

Series[Log[1+(x/n)],{n,∞,5}]

Limit[(1/((Sin[x])^2))-(1/(x^2)),x->0]

(1/((Sin[x])^2))-(1/(x^2))

(1/((Sin[x])^2))-(1/(x^2))==(((x^2)-((Sin[x])^2))/((x^2)((Sin[x])^2)))//Simplify

Block[{nd=4,n=(x^2)-((Sin[x])^2),d=(x^2)((Sin[x])^2)},
Table[
{D[n,{x,i}],D[d,{x,i}]}->({D[n,{x,i}],D[d,{x,i}]}/.x->0),
{i,nd}
]
]

compound[
r_:0.05,
n_:1
]:=
(((1+(r/n)))^n)

1000compound[]

1000compound[0.05,4]

1000compound[0.05,12]

(0.05/4)1000

(1000+(0.05/4)1000)*(0.05/4)

1000(E^0.05)

Series around ∞

Series[(Sqrt[(x^2)+1]),{x,∞,5}]

Series[(E^x),{x,∞,5}]

Series[(1/(Log[x+1])),{x,∞,5}]

Series[(1/(Sin[x])),{x,0,5}]

∞-∞

Limit[(1/x)-(1/(Log[x+1])),x->0]

Block[{nd=4,l=0,v=x,n=Log[x+1]-x,d=x Log[x+1],ds},
ds=Table[
{D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}->({D[n,{v,i}],D[d,{v,i}],D[n,{v,i}]/D[d,{v,i}]}/.v->l),
{i,0,nd}
];
AppendTo[ds,Limit[(n/d),v->l]];
ds
]//Column

(((1/(x+1))-1)/((x/(x+1))+Log[x+1]))//Simplify

D[(x+1)Log[x+1],x]

Limit[((x+Sin[x])/x),x->∞]