If your computer is directly connected to your router (using a lan cable), you could check that the cable is plugged in properly on both sides. Limited connectivity means, that the LAN Card receives a signal through the cable, but the network adapter does not receive an automatically assigned IP Address. This could well be a router problem. Try to restart the internet router and try again.
If that does not work, you could try the following:
On the computer that can go online, click on your start button and select "run". then type cmd into the box and hit enter. continue typing "ipconfig /all" into the black box and hit enter.
It will give you an ip address, a subnetmask, a DNS Server and a gateway.
Write those down.
Then go to the computer that does not work and right click on the icon that sais limited connectivity and select "status". There select "properties" then scroll down the list and select TCP/IP protocoll and hit properties. Now configure the ip settings according to what you wrote down. Duplicate the subnetmask the DNS Server and the gateway from the other computer, but add one number to the ip address. So if the working computer has 192.168.1.100, then use 192.168.1.101 for the other one.
After you configured it, hit ok twice and wait a few seconds. Now try to access the internet. If that works, then the DHCP Server on your router may be misconfigured or faulty.