Here are some Dirac Delta function models, we can see that they peak at x = (Subscript[x, 0]) and will converge to zero everywhere else when applying the proper limit.

f(x)==(Limit2Arg[(α/(π ((α^2)+(x^2)))), α, 0])

Manipulate[Plot[(α/(π*((((x-(Subscript[x, 0])))^2)+(α^2)))),{x,-1,1},PlotRange->All,PlotStyle->{AbsoluteThickness[3.],Black}],{α,.5,0.000001},{{(Subscript[x, 0]),0},-1,1}]

Θ(x)==({{{, ({{1, x>0}, {0, x<0}})}})

ρ(x)==(Limit2Arg[((Θ(x+ϵ)-Θ(x))/ϵ), ϵ, 0])

Θ[x_]:=UnitStep[x];

ρ[x_,ϵ_]:=((Θ[x+ϵ]-Θ[x])/ϵ);

Manipulate[Plot[ρ[x-(Subscript[x, 0]),ϵ],{x,-1,1},PlotStyle->{AbsoluteThickness[3.],Black}],{ϵ,1,0.02},{{(Subscript[x, 0]),0},-.5,1}]

f(x)=(Limit2Arg[((sin(α x))/(π x)), α, ∞])

Manipulate[Plot[((Sin[α*(x-(Subscript[x, 0]))])/(π*(x-(Subscript[x, 0])))),{x,-1,1},PlotStyle->{AbsoluteThickness[3.],Black},PlotRange->All],{α,1,100},{{(Subscript[x, 0]),0},-1,1}]

Ibraheem Al-Yousef, PHYS310.