I'm a bit lost on this problem. I want to show that if a function $f\colon U\subset \mathbb R^n\to \mathbb R$ is of class $C^N$ on $U$ (continuous partial derivatives up to order $N$) and $T_N(\hat{x})$ is the Taylor polynomial of order $N$ at $\hat{x}_0=(x_1^0,\ldots,x_n^0)$ defined by:
\begin{align*} T_N(\hat{x})=f(\hat{x}_0)+\sum_{i=1}^n\frac{\partial f}{\partial x_i}(\hat{x}_0)(x_i-x_i^0)+\cdots+ \frac{1}{N!}\sum_{i_1,\ldots,i_N=1}^n\frac{\partial^N f}{\partial x_{i_N}\cdots\partial x_{i_1}}(\hat{x}_0)(x_{i_1}-x_{i-1}^0)\cdots(x_{i_N}-x_{i_N}^0) \end{align*}
then \begin{align*} \lim_{x\to x_0}\frac{f(\hat{x})-T_N(\hat{x})}{||\hat{x}-\hat{x}_0||^N}=0 \end{align*}
I understand that by making the additional assumption that a function is $C^{N+1}$, we can express the remainder in the Lagrange form. However, if we assume only $C^{N}$, I suspect that we are limited to the Peano form. I'm struggling to find a proof for this. Does anyone have suggestions?
