(*Video 1: Evaluating limits with Mathematica*)

TraditionalForm[HoldForm[Limit[Sqrt[x+3],x->1]]]

Limit[Sqrt[x+3],x->1]

Plot[Sqrt[x+3],{x,-2,2}]

f[x_]:=Sqrt[x+3];
a=-2;b=2;

Manipulate[Grid[{{Row[{"f[t] = ",f[t]}]},{Show[Plot[f[x],{x,a,b}],Plot[f[t],{x,a,b},PlotStyle->Red],ParametricPlot[{t,y},{y,f[a],f[b]},PlotStyle->Gray],ImageSize->500]}},Spacings->{1,1},Frame->All],{{t,1},a,b},TrackedSymbols:>t]

TraditionalForm[HoldForm[Limit[(x^3-1)/(x-1),x->1]]]

Factor[x^3-1]

Simplify[(x^3-1)/(x-1)]

Limit[(x^3-1)/(x-1),x->1]

Clear[a,b,f]
f[x_]:=(x^3-1)/(x-1);
a=0;b=2;
Manipulate[Grid[{{Row[{"f[t] = ",f[t]}]},{Show[Plot[f[x],{x,a,b}],Plot[f[t],{x,a,b},PlotStyle->Red],ParametricPlot[{t,y},{y,f[a],f[b]},PlotStyle->Gray],ImageSize->500]}},Spacings->{1,1},Frame->All],{{t,1},a,b},TrackedSymbols:>t]

Clear[a,b,f]

(*Video 2: Directional limits*)

TraditionalForm[HoldForm[Abs[x]/x]]

Plot[Abs[x]/x,{x,-1,1}]

TraditionalForm[HoldForm[Limit[Abs[x]/x,x->0,Direction->"FromAbove"]]]
TraditionalForm[HoldForm[Limit[Abs[x]/x,x->0,Direction->"FromBelow"]]]

Limit[Abs[x]/x,x->0,Direction->"FromAbove"]

Limit[Abs[x]/x,x->0,Direction->"FromBelow"]

TraditionalForm[HoldForm[Abs[x-2]/(x-2)-x^2+3x]]

Plot[Abs[x-2]/(x-2)-x^2+3x,{x,0,4}]

TraditionalForm[HoldForm[Limit[Abs[x-2]/(x-2)-x^2+3x,x->2,Direction->"FromAbove"]]]

Limit[Abs[x-2]/(x-2)-x^2+3x,x->2,Direction->"FromAbove"]

TraditionalForm[HoldForm[Limit[Abs[x-2]/(x-2)-x^2+3x,x->2,Direction->"FromBelow"]]]

Limit[Abs[x-2]/(x-2)-x^2+3x,x->2,Direction->"FromBelow"]

f[x_]:=Abs[x-2]/(x-2)-x^2+3x;
a=0;b=4;
c=Minimize[{f[x],a≤x≤b},x][[1]];d=Maximize[{f[x],a≤x≤b},x][[1]];

Manipulate[Grid[{{Row[{"f[t] = ",f[t]}]},{Show[Plot[f[x],{x,a,b}],Plot[f[t],{x,a,b},PlotStyle->Red],ParametricPlot[{t,y},{y,c,d},PlotStyle->Gray],ImageSize->500]}},Spacings->{1,1},Frame->All],{{t,2.2},a,b},TrackedSymbols:>t]

Clear[a,b,c,d,f]

(*Video 3: Squeeze Theorem*)

TraditionalForm[HoldForm[x^2 Sin[1/x]]]

Plot[x^2 Sin[1/x],{x,-.1,.1}]

Show[Plot[x^2 Sin[1/x],{x,-.1,.1}],Plot[{x^2,-x^2},{x,-.1,.1},PlotStyle->{Red,Green}]]

Plot[Sin[1/x],{x,-.1,.1}]

(* For all x ≠ 0, we know that x^2>0 *)

(* Therefore, as -1≤Sin[1/x]≤1, we conclude that for all x≠0, 
-x^2 ≤ x^2 Sin[1/x] ≤ x^2
*)

Limit[x^2 ,x->0]
Limit[-x^2 ,x->0]

TraditionalForm[HoldForm[Limit[x^2 Sin[1/x],x->0]]]

TraditionalForm[HoldForm[Limit[-x^2 ,x->0]<=Limit[x^2 Sin[1/x],x->0]<=Limit[x^2 ,x->0]]]

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

(*Try to argue this one on your own: *)

Show[Plot[x Cos[1/x]+1/2,{x,-.1,.1}],Plot[{Abs[x]+1/2,-Abs[x]+1/2},{x,-.1,.1},PlotStyle->{Red,Green}]]

Limit[x Cos[1/x]+1/2,x->0]