A simple example is [nextfloat(0.0)]. In short, there are floating point numbers greater than zero whose inverse is Inf. Multiplying by \frac{εδ}{εδ} can avoid it.
julia> a = [nextfloat(0.0)]
1-element Vector{Float64}:
5.0e-324
julia> nrm =norm(a)
5.0e-324
julia> a*inv(nrm)
1-element Vector{Float64}:
Inf
julia> δ = inv(prevfloat(typemax(nrm)))
5.562684646268003e-309
julia> εδ = eps(one(nrm))/δ
3.99168061906944e292
julia> a*εδ
1-element Vector{Float64}:
1.9721522630525295e-31
julia> ans * inv(nrm*εδ)
1-element Vector{Float64}:
1.0