<s>[INST] For each $a \in \mathbb{R}$ evaluate $ \lim\limits_{n \to \infty}\left(\begin{smallmatrix}1&\frac{a}{n}\\\frac{-a}{n}&1\end{smallmatrix}\right)^n$ 
If $a \in \mathbb{R}$, evaluate $$ \lim_{n \to \infty}\left(\begin{matrix} 1&\frac{a}{n}\\\frac{-a}{n}&1\end{matrix}\right)^{n}$$

My attempt: Let $$A = \left(\begin{matrix} 0&a\\-a&0\end{matrix}\right) = -a\left(\begin{matrix} \cos(\frac{\pi}{2})&-\sin(\frac{\pi}{2})\\\sin(\frac{\pi}{2})&\cos(\frac{\pi}{2})\end{matrix}\right)$$  so that $$A^k = (-a)^k \left(\begin{matrix} \cos(\frac{k\pi}{2})&-\sin(\frac{k\pi}{2})\\\sin(\frac{k\pi}{2})&\cos(\frac{k\pi}{2})\end{matrix}\right)$$
Thus, 
\begin{align}\displaystyle \lim_{n \to \infty}\left(\begin{matrix} 1&\dfrac{a}{n}\\\dfrac{-a}{n}&1\end{matrix}\right)^{n} &=\displaystyle \lim_{n \to \infty} \left(I+\dfrac{A}{n}\right)^n =e^A=\displaystyle \sum_{k=0}^{\infty}\dfrac{A^k}{k!}\\&= \sum_{k=0}^{\infty} \left(\begin{matrix} \dfrac{(-a)^k\cos(\frac{k\pi}{2})}{k!}&-\dfrac{(-a)^k\sin(\frac{k\pi}{2})}{k!}\\\dfrac{(-a)^k\sin(\frac{k\pi}{2})}{k!}&\dfrac{(-a)^k\cos(\frac{k\pi}{2})}{k!}\end{matrix}\right) \end{align}
and since $\displaystyle \sum_{k=0}^{\infty}\dfrac{(-a)^k\cos(\frac{k\pi}{2})}{k!}=1+0-\dfrac{a^2}{2!}+0+\dfrac{a^4}{4!}+\cdots= \cos a$ and
$\displaystyle \sum_{k=0}^{\infty}\dfrac{(-a)^k\sin(\frac{k\pi}{2})}{k!}=0-a+0+\dfrac{a^3}{3!}+0-\dfrac{a^5}{5!}+\cdots= -\sin a$ therefore the required answer is
$\left(\begin{matrix} \cos a&\sin a\\-\sin a&\cos a\end{matrix}\right).$
However the above answer does not match the choices provided which are $I, 0$ and none of the above. So my question is: Is my answer correct?
 [/INST] Your answer is correct as well. but there is a short way to reach it. Merely by putting ourselves in the complex plan where we identify 
$$ 1 \equiv  \left(\begin{matrix}  1& 0\\0&1 \end{matrix}\right)~~~\text{and}~~~~ i \equiv  \left(\begin{matrix}  0& 1\\-1&0 \end{matrix}\right).$$

Thus,  $$\begin{align}\displaystyle \lim_{n \to \infty}\left(\begin{matrix} 1&\dfrac{a}{n}\\\dfrac{-a}{n}&1\end{matrix}\right)^{n} &= \displaystyle \lim_{n \to \infty}\color{blue}{\left(1+\dfrac{ai}{n}\right)^n} 
\\&=\color{red}{e^{ai} = \cos a+i\sin a} \\&= \left(\begin{matrix} \cos a&\sin a\\-\sin a&\cos a\end{matrix}\right).\end{align}$$
Given that, for any $z\in \Bbb C$ we have, $\lim\limits_{n \to \infty}\left(1+\dfrac{z}{n}\right)^n =e^{z} $
See here
also here, 
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