Proof by LH rule: If $f^{(n)}$ exists at $x_0$ and $\lim_{x\to x_0}\frac{f(x)-f(x_0)}{(x-x_0)^n}=0$, is it true that $f^{(k)}(x_0)=0, k=1,2,\cdots,n$?
if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and $\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$ then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$