What do you mean by "libertarian world"? Libertarianism is a spectrum, so it's hard to give examples without knowing what specific kind of libertarian world we are discussing.
For instance, there are some libertarians who say there should be NO government. Everything should be handled by markets and private property rights. To those libertarians, my example would be dealing with crime. For instance, suppose you come home and find that a crime has been committed against you--perhaps something of yours has been stolen, or a family member has been killed.
You call a police company (one of many competing private police companies) and hire them to investigate the crime. As part of their investigation, they determine that they would like to talk to me, and that they believe important evidence is in my house.
When they knock on my door and ask to search my house, I'm going to tell them to go away (and I'm going to back that request with force--both my own guns, and the guns of the private police force I have hired to keep trespassers off my land).
Oops. With no government, who issues search warrants and provides the force to execute them?
I've seen various attempts to resolve this problem, but none that would actually work, where by "work" I mean achieve the outcome that they are intended to achieve.
You realize that's what a government is, right?