$$\lim\limits_{\textbf{x} \rightarrow \textbf{x}_0} \frac{\|f(\textbf{x}) - f(\textbf{x}_0) - \textbf{T}(\textbf{x} - \textbf{x}_0)\|}{\|\textbf{x} - \textbf{x}_0\|} = 0$$
Does the $\textbf{T}$ part mean we evaluate the partials at the point in question ($\textbf{x}_0$)? For example, the function $f(x,y) = xy$ at the point $(1, 1)$. Is it this:
$$\lim\limits_{(x,y) \rightarrow (1,1)} \frac{\|xy - 1 - \textbf{T}(\textbf{x} - \textbf{x}_0)\|}{\sqrt{(x - 1)^2 + (y - 1)^2}} = 0$$
$$\lim\limits_{(x,y) \rightarrow (1,1)} \frac{\|xy - 1 - [y(x - 1) + x(y - 1)]\|}{\sqrt{(x - 1)^2 + (y - 1)^2}} = 0$$
OR
$$\lim\limits_{(x,y) \rightarrow (1,1)} \frac{\|xy - 1 - [1(x - 1) + 1(y - 1)]\|}{\sqrt{(x - 1)^2 + (y - 1)^2}} = 0$$ ?
Let's give it another chance...$T_{(1,1)} = [1,1]$. So $T_{(1,1)}(\textbf{x} - \textbf{x}_0) = 1\cdot (x - 1) + 1\cdot (y - 1) = x + y - 2$, and you are back the the proof I posted earlier.