How to obtain approximate form for a calculation using Mathematica with specific approximations?

Question

I am currently working on a calculation in Mathematica that involves various parameters, including Nb, k, and Ns. I want to obtain an approximate form of the calculation by considering the following approximations: k << 1, Ns << 1 and Nb >> 1.

    ClearAll["Global`*"]
    S = 2 Ns + 1; B = 2 Nb + 1; A = 2*k*Ns + B; Cq = 2*Sqrt[Ns*(Ns + 1)];
    vH01 = B;vH02 = S;
    vH1[k_] = ((-1)^k (S - A) + Sqrt[(A + S)^2 - 4 k*(Cq)^2])/2
    G[s_, x_] = (2^s)/((x + 1)^s - (x - 1)^s)
    G1 = G[1/2, vH01];G2 = G[1/2, vH02];G3 = G[1/2, vH1[1]];G4 = G[1/2, vH1[2]];
        
    L[s_, x_] = ((x + 1)^s + (x - 1)^s)/((x + 1)^s - (x - 1)^s)
    L1 = L[1/2, vH01]; L2 = L[1/2, vH02]; L3 = L[1/2, vH1[1]];L4 = L[1/2, vH1[2]];
        
    xp = Sqrt[0.5 ((A + S) +Sqrt[(A + S)^2 - 4*k*(Cq)^2])/(Sqrt[(A + S)^2 - 4*k* 
    (Cq)^2])]//FullSimplify
    xm = Sqrt[0.5 ((A + S) -Sqrt[(A + S)^2 - 4*k*(Cq)^2])/(Sqrt[(A + S)^2 - 4*k* 
    (Cq)^2])]//FullSimplify
        
    SA ={{xp, 0, xm, 0}, {0, xp, 0, -xm}, {xm, 0, xp,0}, {0, -xm, 0, xp}}
    VAMid = {{L1, 0, 0, 0}, {0, L1, 0, 0}, {0, 0, L2, 0}, {0, 0, 0, L2}}
    VA = (SA).(VAMid).(SA); Vdet = Det[VA]

These are the equations that I am trying to use:

This is the determinant equation that I am trying to simplify: Could you please guide me on how to implement these approximations in Mathematica? Any suggestions, explanations, or code examples would be highly appreciated.

Answer 1

Vdet = Assuming[Nb > 0 && Ns > 0, Det[VA] // FullSimplify]

(* (1 + 2 Nb + 2 Sqrt[Nb (1 + Nb)])^2 (1 + 2 Ns + 2 Sqrt[Ns (1 + Ns)])^2 *)

Let r == Ns/Nb

approx1 = Assuming[Nb > 0 && r > 0,
  FullSimplify[Asymptotic[Vdet /. Ns -> r*Nb, r -> 0]]]

(* 1 + 4 Sqrt[Nb (1 + Nb)] + 8 Nb (1 + Nb + Sqrt[Nb (1 + Nb)]) *)

approx2 = Asymptotic[approx1, {Nb, Infinity, 2}]

(* 2 - 1/(16 Nb^2) + 16 Nb + 16 Nb^2 *)