Apple Opens iPhone 5 Battery Replacement Program For "Small Percentage" Of Defective Units
Ah, but we digress. The point here is that some iPhone 5 models have what Apple finally considers a legitimate problem, and that's the battery. It doesn't affect all handsets, just devices that were sold between September 2012 and January 2013 and that also fall within a specific (and limited) serial number range.
If your iPhone 5 seems to experience battery life woes, you can plug your serial number into Apple's iPhone 5 Battery Replacement Program page to see if you own a defective model. If you do, Apple will replace the battery for you. For that to happen, you have to take the phone to an Apple Authorized Service Provider, Apple Retail Store, or contact Apple Technical Support.
Before taking your phone in for service, Apple asks that you back up your data to iTunes or iCloud, turn off Find my iPhone, and factory reset your device (erase data and settings in Settings > General > Reset > Erase all Content and Settings).