For , we wish to know if has a pole of order greater than one or has a pole of order greater than two. If so, then the singular point is irregular. We can find out by simply taking limits that would cancel a simple pole in the former case or a double pole in the later case. So in the case above with the singular point at x=0, we evaluate the following limits:
If both limits exists, then cannot have a pole of order greater than one, and cannot have a pole greater than two. So just take the limits and see what happens.