I'm trying to express the following function in terms of a limit L plus either big O and little o. In other words,
$$ f(h) = \frac{1+h-e^h}{h^2} $$
should be expressed in the form $f(h) = L+O(h^n) = L+o(h^m)$ as $h \to 0$.
I have evaluated the limit using L'Hopital's rule, and obtain:
$$ \lim_{h \to 0} \frac{1+h-e^h}{h^2} = -\frac{1}{2} $$
However, this does not have any remaining big O or little o term. Another option I explored is to Taylor expand $f(h)$ around $h=0$, but then $O(h^n)$ will be arbitrarily chosen based on where I choose to truncate.
I read an answer to a similar problem here, but it's not clear how the answer was reached, and doesn't say anything about a little o term.
Thanks in advance!