Why do they allow us to use limewire when every download you make is copyright infringement?:
Not every download is copyright infringment.

If someone has legal ownership of a work (as its creator/author), then they are perfectly entitled to grant permission for anyone else to make copies and distribute them.

Also, many works are either public domain or allowed by their copyright holder to be distributed under a specific license agreement (such as non-commercial use only). These are again not in volation of copyright laws.

So, there are plenty of legal ways that software like limewire can be used. And as long as there are legal uses, and as long as the software maker doesn't intend or support illegal use to occur -- the software itself is not illegal.

But using it to make unlawful copies is still illegal.