Some properties of Bessel function of first kind with integer order

Question

I would like to know the derivation of the following properties of Bessel function of first kind $J_{n}(z)$, where $n\in \{-\infty,\dots,-2,-1,0,1,2,\dots,\infty\}$ and $z\in\mathbb{R}$,

1. In this book, page 391 it has been said that for $z \gg 1$ the dominant harmonics of the Bessel function would be closely spaced about $n\sim z$ and for $z\ll 1$ the dominant harmonics would be $n=0,\pm 1,\pm 2$. Is there any derivation for that?

2. Formula for asymptotic expansion of $J_{n}(n\sin(\theta))$ and $J'_{n}(n\sin(\theta))$ in terms of $K_{\mu}$, modified Bessel function.