asymptotics - Solving limits without l'Hopital

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?