Suppose that the power series P n=0 anx n and P n=0 bnx n converge in some non-trivial interval centered at zero, and consider functions f(x) = P n=0 anx n and g(x) = P n=0 bnx n. Prove (without using l'Hopital's Rule!) that if limx0 f(x) = limx0 g(x) = 0, f 0 (0) = g 0 (0) = 0 and g 00(0) = 2023, then limx0 f(x) g(x) = limx0 f 0 (x) g 0(x) .