Yes, it was illegal for the US to intervene in Libya without approval by Congress, because a President can only intervene for a short period of time without asking for Congressional approval. Obama just did not want to officially declare war. But the conflict in Libya lasted too long, and he needed to have asked for Congressional intervention, in order to stay within the confines of the law.
By most precedents, the United States intervention in Libya was illegal because the president didn't seek approval from Congress. Bypassing Congress makes such an act illegal on many grounds, and approval should have been seeked out beforehand. Therefore, the U.S. shouldn't be intervening in Libyan affairs to the degree that it has so far.
It was illegal for the US to intervene in Libya without approval by Congress. I think it goes against the Constitution that the President completely ignored Congress on this decision. However, it was done and we must move on with it. I think though that there are certain procedures in place and they should have been followed.
The US president is able to send forces out for a trial amount of time ( between 30 and 60 days not sure which one ) at that point they retract the troops or congress okays for them to stay. If they are in a situation that makes it hard for them to retreat they get to stay.
No, it was not illegal for the US to intervene in Libya without approval by Congress, because the U.S. Constitution provides for limited military action in some circumstances without Congressional approval. If the military action exceeds a certain period of time the President must ask Congress to authorize the action. But what happened in Libya was brief and was Constitutional.