How to differentiate the function $f(x,y)=(y,x)$ at $ a=(a_1,a_2)$?
$$\begin{align} \lim\limits_{h \to 0}\frac{\Vert f(a+h)-f(a)-Df(a)h\Vert}{\Vert h\Vert} &=\lim\limits_{h \to 0}\frac{\Vert (h_2+a_2,h_1+a_1) -(a_2,a_1)-(0,a_1h_1)-(a_2h_2,0)\Vert}{\Vert h\Vert}\\ &=\lim\limits_{h \to 0}\frac{\Vert (h_2,h_1)-(a_2h_2,a_1h_1)\Vert}{\Vert h\Vert}\\ &=\lim\limits_{h \to 0}\frac{\Vert (h_2(1-a_2),h_1(1-a_1))\vert\vert}{\Vert h\Vert} \end{align} $$
At this point I don't think I can do anything because I don't know what $a_1$ and $a_2$ are. If I knew they were between $[-1,1]$, I could use squeeze theorem but I'm not sure what to do.
Hint: You are trying to linearly approximate the function $$ f(x,y)=(y,x)=\begin{pmatrix} 0&1\\1&0 \end{pmatrix}\begin{pmatrix}x\\y \end{pmatrix} $$