$$ \lim _{x \rightarrow 0} \frac{\tan 2 x-2 \tan x}{(1-\cos x)\left(2^x-1\right)}= $$

$$ \mathop {\lim }\limits_{x \to 0} \frac{\tan ^2\left(\pi \sec ^4 x\right)}{\pi^2 x^4}= $$

$$\mathop {\lim }\limits_{x \to 0}\left(\frac{4!}{x^8}\left(1-\cos \frac{x^2}{3}-\cos \frac{x^2}{4}+\cos \frac{x^2}{3} \cos \frac{x^2}{4}\right)\right)= $$

Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix defined by $a_{i j}=\left\{\begin{array}{cc}k^i, & \forall i=j \\ 0, & \text { otherwise }\end{array}\right.$. If $m=$ trace of $A$ and $\lim _{k \rightarrow 1} \frac{n-m}{1-k}=171$, then the value of $n$ is