We don’t need l’Hôpital’s Rule if we use series for the functions and approximate for c→0⁺.
ln(1+x)≃x when |x|<<1; cosh(x)≃1+½x² for |x|<<1.
Therefore cosh(√(gc/mt))≃1+gc/(2mt).
ln(1+gc/(2mt))≃gc/(2mt).
(m/c)ln(cosh(√(gc/mt))≃(m/c)(gc/(2mt))=½(g/t).
Note that √(gc/mt) requires c≥0 hence c→0⁺ because c cannot be negative.