My router connected to my computer says limited or no connectivity and says cannot establish a internet connection when the internet is working fine

Update your firmware on your router.

Plug a network cable from your PC Network connector directly into your Cable or DSL modem port. Do you have internet? Does it stay up? If no check your PC's network settings or call provider. If yes connect your router into your modem. The cable goes to the WAN or Internet port on your router not a port numbered 1-4. Plug your PC into one of the 1-4 ports. Do you get internet? If no check your routers settings by typing 192.168.1.1 into the address line of a browser. You probably want the default settings which is DHCP and get an IP automatically. If yes try connecting wireless to your router with a phone or tablet. Do this byy going to settings on your device, turning on the wireless and picking your routers name from the list displayed. If you are asked for a password and know what it is enter it. If not you will have to default the router to an open state and try again. Do this by holding down the reset button on the back of the router. So do you have wireless internet? If not check the wireless settings on the router by logging into the router and going to the Wireless tab. Select open or no security until you can hit the internet with your wireless devices, change it later by selecting WPA2 and use your telephone number or something easy to remember. You can change your SSID to a different name or turn off the SSID broadcast if you want and change the password to get in to the router settings. If your mode choices are B or G or both select G only for faster performance. If your choices are B/G/N select auto. Select auto for channel unless you have s/w on a laptop that shows your signal strength. If you do you can set the channel for the best reception in the room you are in most. Move your router away from cement walls, cordless phones, microwaves, CRTs, stereo speakers. Try turning it 90 degrees each way.

May be it's showing limited or no connectivity on your wireless connection where it's trying to connect or may be it's connected wirelessly but your wired connection is not communicating properly with router's switch, could be due to bad port on router or bad NIC or bad cable...check in your network connection.