The cable connection on my tv broke off how can I connect back to the cable
On most TV sets, the cable connector (also called an F-connector) is just press-fitted into the metal housing of the tuner (which selects the channels). It's not very strong mechanically and is pretty easily broken by tripping over the cable, moving the TV if the cable is too short, or often by cable technicians who overtighten the cable connection with a wrench.
The good news is that this can be fixed. There are two ways. The simple method is to replace the tuner. It's quick but costly as a new tuner may run $100. The second way is to replace the broken-off F-connector. On some sets it's possible to do this without removing the tuner, but not always. This repair may take a little more labor time, but saves the cost of the tuner.
Either way, this is a repair that's best left to a professional. Without the soldering skills and equipment needed, it's possible to do major damage to the TV and tuner circuit boards. I would recommend getting an estimate for the repair from a shop in your area. Yes, there are still places that fix TVs, but they are getting scarce.
Another possible "fix" is to forget about using the TV cable connection and use a VCR or DVR to tune the channels, and just use the TV's video and audio inputs. It doesn't repair the TV, but it gets you back onto the cable.
I hope you've found this helpful. Thanks for using Fixya!