My 2007 Caliber was having the same problem as described. The passenger side headlight low-beam did not work despite changing the bulb. I first tried the OBD reset, but a) I didn't have any error codes, and b) it didn't fix my problem.
After another internet search and a couple YouTube videos later, I fixed the problem in about 30 - 45 minutes by running a jumper wire from the driver's side low-beam wire to the passenger side low-beam wire.
Things you need to know:
1) make sure it's not just a bad bulb
2) the passenger side bulb is a bee-awch to work on. The coolant reserve tank and windshield fluid tank are right in the way. The driver's side is much easier, but you have to remove the fresh air cover which is easy-peasy.
3) you'll need to buy some 18G wire and some quick splices to make this go smoothly. AutoZone had what I needed for about $10.
4) you HAVE TO KNOW what wires to splice/jump together. A) The driver's side wire is white with a blue stripe. B) The passenger side wire is violet/purple with a red stripe. Regardless of whether it's the driver's side or passenger side low-beam that's not working, you will be splicing/jumping these two wires.
You'll need wire cutters and needle-nose pliers. Thread the 18G through the front of the car. I tied off the ends to better secure the wire. I found taking the bulbs out massage working on all the wires much easier. (I tried both leaving them in and taking them out) Take you time to ensure that the jumper wire and low-beam wires are secure in the Easy Splices. You really only get 1 good shot at it.
Once your done, test it the lights! Hopefully everything works like mine did.
Images (Click to enlarge)
Saturday, August 11th, 2018 AT 6:35 PM