The Bessel function $J_1$ is the bounded solution of the differential equation Bessel $x^2y''+xy'+(x^2-1)y=0$ at the point $0$ and gets the integral representation $$J_1(x)=\frac{1}{2\pi}\int_{\pi}^{\pi}e^{f(x\sin t-t)}\,dt, \ x\in \mathbb{R}$$
I want to determine the dominant asymptotic behavior of $J_1$ while $x\rightarrow \pm \infty$ and while $x\rightarrow 0$.
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Could you give me a hint for that? Do we have to calcuate the limit of the absolute value of the funtion $J_1$ while $x\rightarrow \pm \infty$ and while $x\rightarrow 0$ ?