Left turn signal beeps very fast and signal light not working, now is working fine and so is signal light

You will find there is no power getting to one indicator bulb. Possibly a bad connection or a broken wire.

Check the left front bulb-sounds like it burned out. Rapid flashing is a sign of a burned bulb, usually.

Hopefully you still have you manual. I will tell you how to reset the HHR, I found it in my manual. Turn the key to on, press lock and unlock at the same time on your key fob. The horn will beep twice telling you that you are in program mode. The left front turn signal light will come on, press the valve core letting air out of the left front tire. The horn will beep when left front is programmed. Then the right front turn signal will come on, let air out till horn beeps. Then right rear, same procedure then left rear same procedure. When that is done horn will beep twice again saying tire program is complete. Turn key off, make sure you air all tires back up.

Sounds like a burnt out bulb, that fast blink is to tell you a bulb needs replaced.

check for a bad bulb. this is how the system alerts you to a non-working light bulb.

has a bulb out on that side

Check the ground wires at the lights are clean and tight.Its a ground problem you have.

Check the front lights. Usually if i flashes fast it indicate a bulb is out on that side.

Your front turn signal socket is shot. Buy a new one at the dealer for about 30.00. It will fix it. You may be able to clean it and wiggle it to get it to work. But it wont last very long. I had the same problem. Replaced the socket and it is fine. Note: If one side is bad the other side it not far behind. Probably 2 or 3 months it will go also. Good Luck.
Bill

This fast flash is caused by a burnt out bulb in the front or rear.