Approximations for the Bessel and Airy functions with an explicit error term

We show how one can obtain an asymptotic expression for some special functions with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function J_ν (x) and the Airy function Ai(x). In particular, we answer a question raised by Olenko and find a sharp bound on the difference between J_ν (x) and its standard asymptotics. We also give a very simple and surprisingly precise approximation for the zeros Ai(x).