I need to compute an asymptotic approximation to this improper integral using the method of stationary phase. $$I(x)=\int_{-\infty}^{0}{({-\xi})^{1.5k}\sin{(\xi{t_{0})}}e^{-i\xi{x}}}d\xi,\:\:\:\text{as }x\to+\infty$$
where $t_{0}$ and $k$ are constants with respect to the variable of integration.
I have attempted this and I am getting a result that seems to diverge probably due to the $-\infty$ limit:
$$I(x)\approx\lim_{a\to-\infty}\left[\frac{(-a)^{1.5k}\sin(at_{0})}{ix}\right]$$ It looks like I am doing something wrong, kindly help.