Proving an expression from Mathematica which is clearly visible from Plots

Question

I have the following Mathematica code:

$Assumptions = {r \[Element] Reals, r >= 0, rh \[Element] Reals, 
   rh > 0, qe \[Element] Reals, qe >= 0, qm \[Element] Reals, qm >= 0,
    z \[Element] Reals, z >= 0, zh \[Element] Reals, zh >= 0};

S[rh_] := \[Pi] rh^2

M[rh_, qe_, qm_] := 
 1/2 Sqrt[\[Pi]/S[rh]] (S[rh]^2/\[Pi]^2 + S[rh]/\[Pi]  + qe^2 + qm^2)

qe = (r* rh* \[Phi]e)/(r - rh);

f[r_, rh_, qm_, \[Phi]e_] := 
 1 + r^2/l^2 - (2 M[rh, qe, qm])/r + (qe^2 + qm^2)/r^2

Veff[r_] = f[r, rh, qm, \[Phi]e]*(L^2/r^2);

r0 = Solve[Simplify[Veff'[r] == 0], r, Reals][[2]];

\[Lambda][rh_, qm_, \[Phi]e_, l_] = 
  1/Sqrt[2] Sqrt[-((r^2 f[r, rh, qm, \[Phi]e])/L^2) Veff''[r]] /. r0 //
    Simplify;

When I try to plot $\lambda$ v/s $r_h$ for specific values of $q_m$ and $\phi_e$, I find that $\lambda$ saturates to the value $\frac{1}{l}$ when I try using different values of $l$ say 1,2,3 etc. To make it concrete, I also try to find the relation of $\lambda$ v/s $l$ graphically by using the following line of code:

\[Lambda]sat[q_, \[Phi]_, l_] = \[Lambda][100, q, \[Phi], l];

Plot[{\[Lambda]sat[1, 1, l], l^-1}, {l, 0, 100}]

As you can visually check that the relation holds from the plot. My only desire is to prove this analytically. I tried using limits and series method on Mathematica but in vain. How to use Mathematica to prove the relation $\lambda_{sat}=1/l$ analytically ?

Answer 1

Printing the expression we see that it is of the form c1 Sqrt[1 + c2/l^2], where c1,c2 are appropriate constants.

Thus, doing some series expansions at small and high l we easily find:

Series[\[Lambda]sat[1, 1, l], {l, 0, 1}]//N//PowerExpand//Rationalize
(* 1./l+7.40445*10^-14 l+O[l]^3 *)

and

Series[\[Lambda]sat[1, 1,l],{l,\[Infinity],2}]//N//PowerExpand//Rationalize
(* 3.84823*10^-7+1.2993*10^6/l^2+O[1/l]^3 *)

In other words, the expression has two regimes:

1. At low-l \[Lambda]sat[1, 1, l]~1/l
2. At high-l \[Lambda]sat[1, 1, l]~b1+b2/l^2, where b1,b2 are some constants given in the series expansion above.

A plot confirms this and shows the transition between the regimes happens at l~10^6: