Limit[
  -m ((Abs[r + 1/2 λ k] - Abs[r - 1/2 λ k])/(Abs[r + 1/2 λ k] Abs[r - 1/2 λ k])),
  λ -> 0]

r = {x, y};
a = {a1, a2};
expr = 1/Sqrt[(r - a).(r - a)] - 1/Sqrt[(r + a).(r + a)];
(Series[expr, {a1, 0, 1}, {a2, 0, 1}] // Normal) /. x^2 -> R^2 - y^2 //
   PowerExpand // Factor

(*  (2 (a1 x + a2 y))/R^3   *)

Block[{r = {x, y, z}, k = {0, 0, 1}},
 Limit[-m ((Norm[r + 1/2 λ k] - Norm[r - 1/2 λ k]) /
         (Norm[r + 1/2 λ k] Norm[r - 1/2 λ k])) /. m -> μ/λ,
       λ -> 0] // Simplify[#, r ∈ Reals] &
 ]
(*  -((z μ)/(x^2 + y^2 + z^2)^(3/2))  *)

Simplify[-((z μ)/(x^2 + y^2 + z^2)^(3/2)),
 Norm[r]^2 == x^2 + y^2 + z^2 && Norm[r] >= 0]
(*  -((z μ)/Norm[r]^3)  *)