= Solution
{c}
On the fixed parameter interval the integrand tends to $\cos\theta$ as $\kappa\to0$ and is bounded by $\cos\theta$. The <dominated convergence theorem>, or <uniform convergence> here, gives
$$
\boxed{\lim_{\kappa\to0}\tau_A=2\sqrt2T\int_0^{\pi/4}\cos\theta\,d\theta=2T.}
$$
This equals Sam's time between departure and reunion. Physically, $dx/dt=-c\kappa\tan\theta$ tends uniformly to zero, so Alice's <worldline> approaches Sam's, and the <time dilation> disappears.
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