confusion with limit theory, involving l'Hopital's rule

This is actually a problem from a subject thats mainly involved with Laplace transforms.

it starts off with expanding f(s) = around infinity (by taylor series)

so firstly, evaluating at s = infinity, which tends towards log (1 + 0 ) = 0

then the next term of the taylor series involves f ' (s) evaluated at infinity

.... right?

f '' (s) =

now for evaluating this one at infinity directly would yield inf / inf = 1 ?

but i know (of) l'Hopital's rule

which i tried

so lim s -> of

= lim s -> inf of

here i derived both top and bottom of the fraction (seperately) with respect to s

long story short, if you use l'hopitals rule for the following terms as well, you get 0 for every term in the taylor series, which im assuming isnt right.

am i using l'hopitals correctly? any other help?

also any tips on using the Math-notation available here? :P