I want to give some justification to the result given above.
Let's look at the quotient of two Bessel functions
$$
C(\alpha,\beta,x)=\frac{J_1(\alpha f(x) )}{J_1(\beta f(x) )}
$$
now assume that $f(x)\sim x^a+iO(x^{a-\delta})$ as $x\rightarrow \infty$ with $a>\delta>0$it is now tempting to just throw away the imaginary part and proceeding with $f\sim x^a$.
The problem is that the Besselfunction with purely real argument has zeros on the real axis so $C(\alpha,\beta,x_{n,\beta})=\infty$ where $x_{n,\beta}$ is defined as $J_1(\beta x_{n,\beta})=0$ .
We therefore induce some singularities which are absent in the original problem by taking only leading order contributions.
This problem can be cured by taking into account the first non vanishing imaginary term, because it will shift away the $x_{n,\beta}$ away from the real axis. We can now safely proceed with
$$
C(\alpha,\beta,x)\sim\frac{J_1(\alpha x^a(1+icx^{a-\delta-1}))}{J_1(\beta x^a (1+icx^{a-\delta-1}) )}
$$
where $c$ is the first coefficient in the asymptotic expansion of the imaginary part of $f(x)$.
Using the asymptotic expansions for the Besselfunction found here , we obtain
$$
C(\alpha,\beta,x)\sim\frac{\sqrt{\beta}}{\sqrt{\alpha}}\times\frac{\cos\left(\alpha x^a(1+icx^{a-\delta-1})-\frac{3 \pi}{4}\right)}{\cos\left(\beta x^a (1+icx^{a-\delta-1}) -\frac{3 \pi}{4}\right)}
$$
Setting $c=\frac{1}{2},a=2,\delta=1$ and calculating
$$
|C(\alpha,\beta,x)|=\sqrt{\Re[C(\alpha,\beta,x)]^2+\Im[C(\alpha,\beta,x)]^2}=\\
\sqrt{\frac{\beta}{\alpha}}
\times \sqrt{ \Im\left(\frac{\cos \left(-\frac{3 \pi }{4}+\alpha x \left(1+\frac{i}{2 x}\right)\right)}{\cos \left(-\frac{3 \pi }{4}+\beta x \left(1+\frac{i}{2 x}\right)\right)}\right)^2+ \Re\left(\frac{\cos \left(-\frac{3 \pi }{4}+ \alpha x \left(1+\frac{i}{2 x}\right)\right)}{\cos \left(-\frac{3 \pi }{4}+\beta x \left(1+\frac{i}{2 x}\right)\right)}\right)^2}
$$
As stated in the comments follows.
Note that this result doesn't decay to zero or grows to infinity as $x\rightarrow \infty$ as the leading term is of oscillatory nature.
Edit:
I am quite confident that the error is of order $\mathcal{O}(\frac{1}{x})$
Edit2:
![Comparison](https://cdn.statically.io/img/i.sstatic.net/lC1Xx.jpg)
To give an impression how good this approximation is, i plotted
$
\color{red}{|C(2,1,x)|} \quad \text{and} \quad \color{blue}{|C(1,2,x)|}
$
Dotted lines are asymptotics, full lines the exact expressions
Edit3:
One may also notice that this method works for all $J_N(x)$ but the convergence gets worse because sub-leading terms are enhanced by $4 N^2$