algebraic manipulation of probability generating functions

I am revising for a statistics test and I have an urgent problem involving probability generating functions. I understand the concept its just the algebraic manipulation that gets me. My exam is only a few days away so if you could help me out that would be GREAT! :)

I'm supposed to determine the probability distribution of X from the pgf below.

Here is the working I have done so far:

for ,

therefore

This matches exactly with what the lecturer had done exept I couldn't understand how he leapt to this next part:

*and* for x=>1,

Could someone please explain to me how he got this?