integration - Asymptotic behaviour of the propagator for a scalar field

When discussing causality in Chapter 2 of Peskin & Schroeder a couple of equations giving the asymptotic behaviour of the propagator for a scalar field appear:

$$\text{If} \,\, x^0-y^0=t, \, \, \mathbf{x-y} = 0 \Rightarrow D (x-y) = \frac{1}{4 \pi^2} \int_{m}^{\infty} dE \sqrt{E^2-m^2} e^{-iEt} \underset{t \to \infty}{\sim} e^{-imt}$$ $$\text{If} \,\, x^0-y^0=0, \, \, \mathbf{x-y} = \mathbf{r} \Rightarrow D (x-y) = \frac{1}{4 \pi^2 r} \int_{m}^{\infty} d\rho \frac{\rho\, e^{-\rho r}}{\sqrt{\rho^2 - m^2}} \underset{r \to \infty}{\sim} e^{-mr}$$

I can't see how you derive these asymptotic behaviours (I have no problem deriving the integral exact expressions, but then I get stuck). All I could do was to rewrite the first integral as follows:

$$D (x-y) = \frac{1}{4 \pi^2} \int_{m}^{\infty} dE \sqrt{E^2-m^2} e^{-iEt} = \frac{m}{4 \pi^2 i t} K_1(imt)$$

using this article on modified Bessel functions of the second kind. But checking with Mathematica, this vanishes for $t \to \infty$. For the second integral I don't have any clue, so any help would be more than welcome!

Extra (but related) question:

In the first discussion of the chapter something similar appears

$$U(t) = \frac{1}{2 \pi^2 | \mathbf{x - x_0} | } \int_{0}^{\infty}dp\,p\, \sin (p | \mathbf{x - x_0} | ) e^{-it \sqrt{p^2 + m^2}}$$

Is this also obtained through a similar procedure?