How to find the asymptotic expansion of $\int_{-\infty}^{y} e^{-x^2/2}/\sqrt{2\pi} dx$ where $x \in N(0,1)$?

Question

I realize the function inside the integral is the pdf of a normally distributed random variable x, but am unsure how to use this to solve the problem.

I am trying to relate it to the inverse of the taylor series expansion with 1/x instead of x. With x the series will blow up, but with 1/x it has a limiting distribution.

This is as far as I could conclude.