Context: I am preparing the next version of my notes for 2018-2019 and I am including a new section about Bessel functions of the first kind, Voronoi summation and the Gauss circle problem, including some answers I gave on MSE and some material from Watson's Treatise on the Theory of Bessel functions. I was wondering about an alternative proof of Tricomi's inequality
$$ J_0(x) \approx \frac{\sin(x)+\cos(x)}{\sqrt{\pi x}}=\sqrt{\frac{2}{\pi}}\left(J_{1/2}(x)+J_{-1/2}(x)\right)\quad\text{for }x\gg 1 $$
and I realized that something strange is happening here and I don't know what it is.
The Laplace transform of $J_0(x)$ is simply $\frac{1}{\sqrt{1+s^2}}$, and over $\mathbb{R}^+$ the function $a(s)=\frac{1}{1+s^2/2}$ provides a better approximation than $b(s)=\sqrt{\sqrt{1+s^2}-s}+\sqrt{\frac{\sqrt{1+s^2}+s}{1+s^2}}-1$ for $(\mathcal{L} J_0)(s)$, but for large values of $x$ the Bessel function $J_0(x)$ is much closer to $(\mathcal{L}^{-1}b)(x)=\sqrt{\frac{2}{\pi}}\left(J_{1/2}(x)+J_{-1/2}(x)\right)$ than to $(\mathcal{L}^{-1}a)(x)=\sqrt{2}\sin(x\sqrt{2})$. Why?