lim c to 0 from the right, (m/c(ln(cosh sqrt(gc/mt))))

Question

I know you have to do L'Hopitals twice, and the answer is g/2t but have no idea how to do it. Please help

Answer 1

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.