How do you fix limited connection issue?

I have a router and it works fine when my internet is hooked up to it. Well not necessarily fine my internet has been giving me problems, however its connected; there is no "limited connection" issue. Now that i want to check if its the modem or router i hook up the computer directly to the modem and it says limited internet connection. how do i fix this to check? and will i have the problem switching back to the router?