Rules when calculating with limits

My problem is inspired by If $g$ is continuous and $f(a)=0$ then $fcdot g$ is differentiable at $a$ .

Let’s assume that $g:mathbb{R}^mto mathbb{R}^n $ is continuous at $a$, $f:mathbb{R}^mto mathbb{R}^n$ is differentiable at $a$ and $f(a)=0$. Let be $A$ a $mtimes n$ matrix with the property $limlimits_{xto a}frac{f(x)-f(a)-A(x-a)}{Vert x-aVert}=0$. Now consider the following limit:
$$
limlimits_{xto a}frac{f(x)g(x)-f(a)g(a)-Ag(a)(x-a)}{Vert x-aVert}=limlimits_{xto a}frac{f(x)g(x)-Ag(a)(x-a)}{Vert x-aVert}=cdots
$$
I am not sure if I can proceed with following manipulations:
$$cdots=limlimits_{xto a}frac{f(x)}{Vert x-aVert}g(limlimits_{xto a}x)-limlimits_{xto a}frac{Ag(a)(x-a)}{Vert x-aVert}=
limlimits_{xto a}frac{f(x)}{Vert x-aVert}g(a)-limlimits_{xto a}frac{Ag(a)(x-a)}{Vert x-aVert}=\left(limlimits_{xto a}frac{f(x)}{Vert x-aVert}-limlimits_{xto a}frac{A(x-a)}{Vert x-aVert}right)g(a)=limlimits_{xto a}frac{f(x)-A(x-a)}{Vert x-aVert}g(a)=0.
$$

It feels wrong but I am not sure at which step I have violated any limit rule?