Let $\displaystyle I \subseteq \mathbb{R}$ be an interval and let $\displaystyle c \in I$. Suppose that $\displaystyle f$ and $\displaystyle g$ are defined on $\displaystyle I$ and that the derivatives $\displaystyle f^{(n)}$,$\displaystyle g^{(n)}$ exist and are continuous on $\displaystyle I$. If $\displaystyle f^{(k)}(c)=0$ and $\displaystyle g^{(k)}(c)=0$ for $\displaystyle k=0,1,...,n-1$, but $\displaystyle g^{(n)}(c) \neq 0$, show that

$\displaystyle \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{f^{(n)}(c)}{g^{(n)}(c)}$