Function Limits

This notebook explores how functions may or may not approach a limit.

f0[x_]:=(x+1)/x;
Plot[f0[x],{x,-10,10},AxesOrigin->{0,0},PlotStyle->Thick];
Expand[f0[x]];

Plot[{f0[x],1},{x,-20,20},AxesOrigin->{0,0},PlotStyle->Thick];
Limit[f0[x]-1,x->Infinity];

f1[x_]:=( 2x^3+5x^2)/(x^2+3);
Plot[f1[x],{x,-10,10},AxesOrigin->{0,0},PlotStyle->Thick];
Expand[f1[x]];

Plot[{f1[x],2x+5},{x,-20,20},AxesOrigin->{0,0},PlotStyle->Thick];
Limit[f1[x]-(2x+5),x->Infinity];

f2[x_]:=( 2x^4+5x^2)/(x^2+3);
Plot[f2[x],{x,-10,10},AxesOrigin->{0,0},PlotStyle->Thick];
Expand[f2[x]];

Plot[{f2[x],2x^2-1},{x,-3,3},AxesOrigin->{0,0},PlotStyle->Thick];
Limit[(f2[x]-(2x^2-1)),x->Infinity];

f3[x_]:=Sign[x];
Plot[f3[x],{x,-10,10},AxesOrigin->{0,0},PlotStyle->Thick];

(* not correct *)

f4[x_]:=Sin[x]/x;
Plot[f4[x],{x,-12,12},AxesOrigin->{0,0},PlotStyle->Thick];

Limit[(f4[x]),x->0];

f4[x_]:=Sin[x]*x;
Plot[f4[x],{x,-20,20},AxesOrigin->{0,0},PlotStyle->Thick];

Limit[(f4[x]),x->0];

f5[x_]:=Sin[1/x];
Plot[f5[x],{x,-2,2},AxesOrigin->{0,0},PlotStyle->Thick];

Limit[(f5[x]),x->0];

f6[x_]:=x*Sin[1/x];
Plot[f6[x],{x,-2,2},AxesOrigin->{0,0},PlotStyle->Thick];

Limit[(f6[x]),x->0];
Limit[(f6[x]),x->Infinity];

f7[x_]:=Floor[x];
Plot[f7[x],{x,-5,5},AxesOrigin->{0,0},PlotStyle->Thick]

f8[x_]:=Floor[1/x]
(*f8[x_]:=1/Floor[1/x];*)
Plot[f8[x],{x,-1,1},AxesOrigin->{0,0},PlotStyle->Thick]

f9[x_]:=If[MemberQ[Rationals,x],1,0];
Plot[f9[x],{x,-2,2},AxesOrigin->{0,0},PlotStyle->Thick];