(*Video 1: Continuity*)
TraditionalForm[HoldForm[f[x]=Piecewise[{{x^2+1,x<0},{x+c,x>=0}}]]]

Manipulate[Plot[Piecewise[{{x^2+1,x<0},{x+c,x>0}}],{x,-3,3},PlotRange->{-4,4},Exclusions->x==0],{c,-3,3}]

Limit[x^2+1,x->0,Direction->"FromBelow"]
Limit[x+c,x->0,Direction->"FromAbove"]

(*These lateral limits match if and only if c=1. Thus, c=1 is the only value that makes f[x] continuous at x=0.*)

TraditionalForm[HoldForm[f[x]=Piecewise[{{c x^2+2c x+1,x<1},{c x-2c,x>=1}}]]]
Manipulate[Plot[Piecewise[{{c x^2+2c x+1,x<1},{c x-2c,x>=1}}],{x,-5,5},PlotRange->{-6,6},Exclusions->x==1],{c,-3,3}]

Limit[c x^2+2c x+1,x->1,Direction->"FromBelow"]
Limit[c x-2c,x->1,Direction->"FromAbove"]

Solve[1+3c==-c,c]

(*The unique value of c such that f[x] is continuous at x=1 is c=-1/4.*)

(*Video 2: Infinite limits*)

TraditionalForm[HoldForm[Limit[(x^3+2x+1)/(2x^3-5x+3),x->Infinity]]]

(*Numerator*)
x^3(1+2/x^2+1/x^3)
Simplify[x^3(1+2/x^2+1/x^3)]

(*Denominator*)
x^3(2-5/x^2+3/x^3)
Simplify[x^3(2-5/x^2+3/x^3)]

TraditionalForm[HoldForm[x^3(1+2/x^2+1/x^3)/(x^3(2-5/x^2+3/x^3))]]
TraditionalForm[HoldForm[(1+2/x^2+1/x^3)/((2-5/x^2+3/x^3))]]

Limit[(x^3+2x+1)/(2x^3-5x+3),x->Infinity]

Plot[{(x^3+2x+1)/(2x^3-5x+3),1/2},{x,1,100},PlotRange->{0,1}]

(*y=1/2 is a horizontal asymptote for this function as x-->Infinity.*)

TraditionalForm[HoldForm[Limit[(x^3+2x+1)/(x^5-2x+1),x->Infinity]]]

Limit[(x^3+2x+1)/(x^5-2x+1),x->Infinity]

Plot[{(x^3+2x+1)/(x^5-2x+1),0},{x,1,30},PlotRange->{-1,1}]

(*y=0 is a horizontal asymptote for this function as x-> Infinity.*)

TraditionalForm[HoldForm[Limit[(x^6+x+10)/(x^4-2x+1),x->Infinity]]]

Limit[(x^6+x+10)/(x^4-2x+1),x->Infinity]

Plot[(x^6+x+10)/(x^4-2x+1),{x,1,1000}]

(*Video 3: Asymptotes*)

graph=Plot[1/(x-3),{x,-4,10},PlotRange->10]

Limit[1/(x-3),x->3,Direction->"FromBelow"]
Limit[1/(x-3),x->3,Direction->"FromAbove"]

Vasymp=ParametricPlot[{3,y},{y,-10,10},PlotStyle->Red];(* x=3 is a vertical asymptote for this function*)
Hasymp=ParametricPlot[{x,0},{x,-10,10},PlotStyle->Green];
(* y=0 is a horizontal asymptote for this function*)
Show[graph,Vasymp,Hasymp]

Manipulate[Plot[1/(x-a)+b,{x,-4,10},PlotRange->10,Exclusions->x==a],{a,0,5},{b,-5,5}]

Plot[{1/2+1/(x^2+1),1/2},{x,-60,60},PlotRange->3]

Limit[1/2+1/(x^2+1),x->Infinity]
Limit[1/2+1/(x^2+1),x->-Infinity]

(*y=1/2 is a horizontal asymptote for this function.*)

TraditionalForm[HoldForm[x/3+5/(x^2+3)+10]]

Limit[x/3+5/(x^2+3)+10,x->Infinity]

Plot[{x/3+5/(x^2+3)+10,x/3+10},{x,-50,50},PlotRange->All]

Limit[(x/3+5/(x^2+3)+10)-(x/3+10),x->Infinity]
Limit[(x/3+5/(x^2+3)+10)-(x/3+10),x->-Infinity]

(* y = x/3+10 is a (slanted) asymptote for this function: not horizontal, nor vertical.*)