Limits with a jump on f[x]=Cos[x]+Sign[Cos[x]].

f[x_]=Cos[x]+Sign[Cos[x]];
a=(π/2);

lim01[h_,LR_,PT_]:=Show[{Plot[f[x],{x,0,(π/2)-0.025}],Plot[f[x],{x,(π/2)+0.025,π}],Graphics[{{Red,Disk[{a-LR(1.5-(h/20)),f[a-LR(1.5-(h/20))]},0.05]},{If[PT==1,White,Black],Disk[{(π/2),PT},0.05]},Circle[{(π/2),1},0.05],Circle[{(π/2),-1},0.05]}]},PlotRange->{{-0.25,π+0.3},{-2.25,2.25}},Ticks->{{{a-1.5+(h/20),"",.02},0,(π/4),(π/2),((3π)/4),π},{{f[a-1.5+(h/20)],"",.02},-2,-1,0,1,2}},AspectRatio->1,AxesLabel->{x,y},AxesOrigin->{0,0}];

Manipulate[
Dynamic[LR];LR=If[Side=="Left",1,-1];
Dynamic[PT];PT=If[Include=="Left",f[(π/2)-0.0025],If[Include=="Floating",0.5,If[Include=="Right",f[(π/2)+0.0025],1]]];
lim01[h-1.01,LR,PT],

{{h,1,"x"},1,31},
{Side,{"Left","Right"}},
{{Include,"Floating","Point"},{"Left","Floating","Right","None"}}
]

Limits with at both infinities on g[x]=(1/(10(((x-(π/2)))^1))).

g[x_]=(1/(10(((x-(π/2)))^1)));
a=(π/2);

lim02[h_,LR_]:=Show[{Plot[g[x],{x,0,(π/2)},PlotRange->{-2.25,2.25}],Plot[g[x],{x,(π/2),π},PlotRange->{-2.25,2.25}],Graphics[{{Red,Disk[{a-LR(1.5-(h/20)),g[a-LR(1.5-(h/20))]},0.05]},{{Dashing[{0.025}],RGBColor[0,0,0],Line[{{(π/2),-2.25},{(π/2),2.25}}]}}}]},PlotRange->{{-0.25,π+0.3},{-2.25,2.25}},Ticks->{{{a-LR(1.5-(h/20)),"",.02},0,(π/4),{(π/2)+0.15,"(π/2)",0},((3π)/4),π},{{g[a-LR(1.5-(h/20))],"",.02},-2,-1,0,1,2}},AspectRatio->1,AxesLabel->{x,y}];

Manipulate[
Dynamic[LR];LR=If[Side=="Left",1,-1];
lim02[h-1.01,LR],

{{h,1,"x"},1,31},
{Side,{"Left","Right"}}
]

Limits with at infinities on j[x]=(1/(30(((x-(π/2)))^2))).

j[x_]=(1/(30(((x-(π/2)))^2)));
a=(π/2);

lim03[h_,LR_]:=Show[{Plot[j[x],{x,0,(π/2)},PlotRange->{-2.25,2.25}],Plot[j[x],{x,(π/2),π},PlotRange->{-2.25,2.25}],Graphics[{{Red,Disk[{a-LR(1.5-(h/20)),j[a-LR(1.5-(h/20))]},0.05]},{{Dashing[{0.025}],RGBColor[0,0,0],Line[{{(π/2),-2.25},{(π/2),2.25}}]}}}]},PlotRange->{{-0.25,π+0.3},{-2.25,2.25}},Ticks->{{{a-LR(1.5-(h/20)),"",.02},0,(π/4),{(π/2)+0.15,"(π/2)",0},((3π)/4),π},{{j[a-LR(1.5-(h/20))],"",.02},-2,-1,0,1,2}},AspectRatio->1,AxesLabel->{x,y}];

Manipulate[
Dynamic[LR];LR=If[Side=="Left",1,-1];
lim03[h-1.01,LR],

{{h,1,"x"},1,31},
{Side,{"Left","Right"}}
]

Limits on a smooth function with a hole: k[x]=(((x-(π/2)))/((x-(π/2))(x+(π/2)))).

k[x_]=(((x-(π/2)))/((x-(π/2))(x+(π/2))));
a=(π/2);



lim04[h_,LR_,PT_]:=Show[{Plot[k[x],{x,0,(π/2)-0.035},PlotRange->{-2.25,2.25}],Plot[k[x],{x,(π/2)+0.035,π},PlotRange->{-2.25,2.25}],Graphics[{{Red,Disk[{a-LR(1.5-(h/20)),k[a-LR(1.5-(h/20))]},0.05]},{PT,Disk[{(π/2),k[(π/2)-0.001]},0.05]},Circle[{(π/2),k[(π/2)-0.001]},0.05]}]},PlotRange->{{-0.25,π+0.3},{-2.25,2.25}},Ticks->{{{a-LR(1.5-(h/20)),"",.02},0,(π/4),{(π/2)+0.15,"(π/2)",0},((3π)/4),π},{{k[a-LR(1.5-(h/20))],"",.02},-2,-1,0,1,2}},AspectRatio->1,AxesLabel->{x,y}];

Manipulate[
Dynamic[LR];LR=If[Side=="Left",1,-1];
Dynamic[PT];PT=If[Include=="No",White,Black];
lim04[h-1.01,LR,PT],

{{h,1,"x"},1,31},
{Side,{"Left","Right"}},
{Include,{"No","Yes"}}
]