<s>[INST] Finding matrix $A^n,$ when $\lim_{n \to \infty}$ 
Finding $\lim_{n\rightarrow \infty}\begin{pmatrix}
1 & \frac{x}{n}\\ \\
-\frac{x}{n} & 1 
\end{pmatrix}^n$ for all $x\in \mathbb{R}$

Try:
 Let $$ A = \begin{pmatrix}1&\frac{x}{n}\\\\-\frac{x}{n}&1\end{pmatrix}.$$
Then $$ A^2 = \begin{pmatrix}1-\frac{x^2}{n^2}&\frac{2x}{n}\\\\-\frac{2x}{n}&1-\frac{x^2}{n^2}\end{pmatrix}$$
And then $$A^3 = \begin{pmatrix}1-3\frac{x^2}{n^2}&\frac{3x}{n}-\frac{x^3}{n^3}\\\\-3\frac{x}{n}+\frac{x^3}{n^3}&1-\frac{x^2}{n^2}\end{pmatrix}$$
So by using same way and taking $\lim_{n\rightarrow \infty}A^n = \begin{pmatrix}1&0\\\\0&1\end{pmatrix}$
But answer given as $$\begin{pmatrix}\cos x &\sin x\\\\-\sin x&\cos x\end{pmatrix}$$
Could some help me where I am missing and also explain how  to solve it?
 [/INST] You seem to be arguing that the terms proportional to $x$, $x^2$, $x^3$, etc. vanish in the limit $n \to \infty$.   But the problem with this argument is that the numerical coefficients in front of these terms also grow as $n \to \infty$, and diverge without bound.  To see this, define
$$
A_n = \begin{bmatrix}1 & x/n\\ -x/n & 1 \end{bmatrix}^n. $$
Here are the terms in the upper-left entry for the first few values of $n$:
\begin{align}
(A_1)_{12} &= \frac{x}{n} = x \\
(A_2)_{12} &= \frac{2x}{n} = x\\
(A_3)_{12} &= \frac{3x}{n} - \frac{x^3}{n^3} = x - \frac{x^3}{27} \\
(A_4)_{12} &= \frac{4 x}{n} - \frac{4 x^3}{n^3} = x - \frac{x^3}{16} \\
(A_5)_{12} &= \frac{5 x}{n} - \frac{10 x^3}{n^3} + \frac{x^5}{n^5} = x - \frac{10 x^3}{125} + \frac{x^5}{3125}\\
\end{align}
Just looking at the term proportional to $x$ in this sequence, it seems pretty likely that this will not vanish as $n \to \infty$.
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