I've encountered several limits that need to be solved in order to calculate the coefficients in asymptotic formulas for elementary functions, e.g.:
$$ \lim_{x \to 0}\frac{\sin x -x}{x^3} $$
$$ \lim_{x \to 0}\frac{e^x-1-x}{x^2} $$
Obviously, they're really easy with l'Hopital, but it's illegal since we haven't reached derivatives yet. Are they possible to solve without l'Hopital? Has anybody encountered the same task with asymptotic formulas, i.e.: finding coefficients for higher order asymptotic decompositions while knowing lower order coefficients?
I used numerical approximation. For the first of your limits, since $x \rightarrow 0$, I substituted $x=.1,.01,.001,.0001$ etc. The answer seemed to be limiting towards $-0.1666$ repeating, which is $x=-1/6$. I assume the same can be done for the other.