= Solution
The one-dimensional <Lorentz transformation> gives
$$
ct'=\gamma(ct-\beta x),
\qquad
x'=\gamma(x-\beta ct),
\qquad
\gamma=\frac1{\sqrt{1-\beta^2}}.
$$
Hence
$$
\boxed{
A_{\mathrm{SR}}
=\gamma
\begin{pmatrix}
1&-\beta\\
-\beta&1
\end{pmatrix}}.
$$
As $|\beta|\to0$, $\gamma=1+O(\beta^2)$. The spatial transformation is therefore $x'=x-ut+O(\beta^2)$, while
$$
t'=t-\frac{ux}{c^2}+O(\beta^2t).
$$
Under the stated condition $|x|<c|t|$, the fractional correction $|ux|/(c^2|t|)$ is smaller than $|\beta|$ and tends to zero. Thus the Lorentz transformation has the <nonrelativistic limit> given by the Galilean transformation.
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