P3365
Let $\varepsilon > 0$. If $x, x_0 \in \mathbb{R}^{N \times 1}$ such that $x \neq x_0$, then we have
\begin{equation}
\begin{split}
\frac{|f(x) - f(x_0) - L(x - x_0)|}{|x - x_0|}
& = \frac{|a^T x + b - (a^T x_0 + b) - a^T (x - x_0)|}{|x - x_0|} \\
& = \frac{|a^T x + b - a^T x_0 - b - a^T x + a^T x_0)|}{|x - x_0|} \\
& = \frac{|0|}{|x - x_0|} \\
& = 0 \\
& < \varepsilon
\end{split}
\end{equation}
Since $\varepsilon > 0$ was arbitrary, the claim follows. $\square$