What is exactly the asymptotic form of Bessel function?

Question

My differential equations book says that $$J_0(x)\simeq\sqrt{\frac{2}{\pi x}}\cos\left(x-\frac{\pi}{4}\right)$$ for "large $x$." It calls it an "asymptotic approximation." My natural questions are: It is an approximation in what sense? Does their difference go to zero as $x\to\infty$ ? Or does their ratio go to 1 as $x\to\infty$ ? How large should $x$ be in order to reduce the error within a specific level? Or is it even possible?