Maple gave me this... $$ J_0(x) = \left( {\frac {\sin \left( x \right) }{\sqrt {\pi}}}+{\frac {\cos \left( x \right) }{\sqrt {\pi}}} \right) x^{-1/2}+ \left( {\frac {\cos \left( x \right) }{8\sqrt {\pi}}}+{\frac {\sin \left( x \right) }{8\sqrt {\pi}}} \right) x^{-3/2} \\+ \left( -{\frac {9\,\sin \left( x \right) }{128\,\sqrt {\pi}}}-{ \frac {9\,\cos \left( x \right) }{128\,\sqrt {\pi}}} \right) x^{-5/2}+ \left( {\frac {75\,\cos \left( x \right) }{ 1024\,\sqrt {\pi}}}-{\frac {75\,\sin \left( x \right) }{1024\,\sqrt { \pi}}} \right) x^{-7/2} \\+ \left( {\frac {3675\, \sin \left( x \right) }{32768\,\sqrt {\pi}}}+{\frac {3675\,\cos \left( x \right) }{32768\,\sqrt {\pi}}} \right) x^{-9/2} \\+ \left( -{\frac {59535\,\cos \left( x \right) }{262144 \,\sqrt {\pi}}}+{\frac {59535\,\sin \left( x \right) }{262144\,\sqrt { \pi}}} \right) x^{-11/2}+o(x^{-11/2}) $$ as $x \to \infty$

Maple gave me this... $$ J_0(x) = \left( {\frac {\sin \left( x \right) }{\sqrt {\pi}}}+{\frac {\cos \left( x \right) }{\sqrt {\pi}}} \right) x^{-1/2}+ \left( -{\frac {\cos \left( x \right) }{8\sqrt {\pi}}}+{\frac {\sin \left( x \right) }{8\sqrt {\pi}}} \right) x^{-3/2} \\+ \left( -{\frac {9\,\sin \left( x \right) }{128\,\sqrt {\pi}}}-{ \frac {9\,\cos \left( x \right) }{128\,\sqrt {\pi}}} \right) x^{-5/2}+ \left( {\frac {75\,\cos \left( x \right) }{ 1024\,\sqrt {\pi}}}-{\frac {75\,\sin \left( x \right) }{1024\,\sqrt { \pi}}} \right) x^{-7/2} \\+ \left( {\frac {3675\, \sin \left( x \right) }{32768\,\sqrt {\pi}}}+{\frac {3675\,\cos \left( x \right) }{32768\,\sqrt {\pi}}} \right) x^{-9/2} \\+ \left( -{\frac {59535\,\cos \left( x \right) }{262144 \,\sqrt {\pi}}}+{\frac {59535\,\sin \left( x \right) }{262144\,\sqrt { \pi}}} \right) x^{-11/2}+o(x^{-11/2}) $$ as $x \to \infty$