= Solution
Apply the <stationary phase method> to the real part of the exponential integral for the <Bessel function of the first kind>. With $n$ fixed, use <oscillatory phase> $\sin\vartheta$ and <oscillatory integral amplitude> $e^{-in\vartheta}$. The unique <stationary point> is $\vartheta_0=\pi/2$, with second <derivative> $-1$. Its contribution is
$$
\frac1\pi\Re\left[e^{-in\pi/2}e^{ix-i\pi/4}\sqrt{\frac{2\pi}{x}}\right].
$$
The leading endpoint terms of the exponential integral are imaginary and do not contribute to its real part. Hence the <large-argument asymptotic expansion of the Bessel function of the first kind> is
$$
\boxed{J_n(x)=\sqrt{\frac{2}{\pi x}}\cos\left(x-\frac{n\pi}{2}-\frac\pi4\right)+O(x^{-3/2})}.
$$
This is an additive asymptotic formula: a relative ratio is inappropriate at zeros of the leading cosine. The envelope decreases like $x^{-1/2}$ while the oscillation <frequency> approaches one.
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