My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$
The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever the functions $F$ and $G$ have continuous partial derivatives up to order two. To show this , we use Taylor expansions about the point $(x_0,y_0)$ to write $F(x,y)$ and $G(x,y)$ in the form $$F(x,y)=F(x_0,y_0)+F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)+\eta_1(x,y)\\ G(x,y)=G(x_0,y_0)+G_x(x_0,y_0)(x-x_0)+G_y(x_0,y_0)(y-y_0)+\eta_2(x,y)$$ where $\eta_1(x,y)/[(x-x_0)^2+(y-y_0)^2]^{1/2}\to 0$ as $(x,y)\to(x_0,y_0)$, and similarly for $\eta_2$.
By locally linear, the book means that the nonlinear system of differential equations around the critical point can be approximated by the linear system $$\mathbf{u}^\prime=\left( \begin{array}{@{}cc@{}} F_x(x_0,y_0)&F_y(x_0,y_0)\\ G_x(x_0,y_0)&G_y(x_0,y_0) \end{array} \right)\mathbf{u}$$ where $\mathbf{u}=\mathbf{x}-\mathbf{x}^0$. The critical point $\mathbf{x}^0=(x_0,y_0)$ is an isolated critical point where $$F(x_0,y_0)=0\quad G(x_0,y_0)=0\\ \det\left(\begin{array}{@{}cc@{}} F_x(x_0,y_0)&F_y(x_0,y_0)\\ G_x(x_0,y_0)&G_y(x_0,y_0) \end{array}\right)\neq0$$
Why does $F$ and $G$ need to have second order partial derivatives if the first order partial derivatives only show up in this expression? And how does that guarantee that $\mathbf{\eta} /||\mathbf{x}-\mathbf{x}^0||$ will go to $0$ as $\mathbf{x}$ approaches the critical point?