A couple of possibilities:
1) Allot more memory to the LimeWire application.
Quit LimeWire. Then click (once) on the LimeWire application
(the original, not an alias) to highlight it. Then go to File
Get Info and choose Memory. Increase it to 10000 KB (at
least). Try to start it again and see what happens.
2) Disable the Fax STF extensions. They conflict with LimeWire.
Look in your Extensions folder. Find any Fax extensions and
move them into your Extensions Disabled folder. (If you don't
have one, then create one.) Or use your Extensions Manager.
I'm not positive if either of these are the cause in this case. But, both can cause unexplained quitting.