Combine two Taylor expansions (Lindeberg CLT)

Question

I am currently reading a proof of Lindeberg-Feller CLT from this article. Here's the problem I encountered:

I can understand the idea of using $x^3$ term to estimate small $x$ and use $x^2$ to estimate large $x$. But the problem is, why does the last equality hold? I try to do some calculations and cannot find an answer.

Also, most proof of the CLT use the estimation like this

and I can't understand why this holds either.