If you were an early purchaser of the iPhone 5 like I was, you may be eligible for a free replacement battery from Apple. As noted on this page of the Apple website: "Apple has determined that a very small percentage of iPhone 5 devices may suddenly experience shorter battery life or need to be charged more frequently. The affected iPhone 5 devices were sold between September 2012 and January 2013 and fall within a limited serial number range." I gave my iPhone 5 to my wife last year when I upgraded to an iPhone 5s. She uses it every day and tells me that — so far — she hasn't noticed shorter battery life, but it turns out that the iPhone 5 I bought on September 21, 2012 and that she now uses is a part of the serial number range.
To find out if your iPhone 5 is affected, determine your iPhone's serial number by opening up the Settings app and then tapping General and then About. You will see a list of lots of information about your iPhone 5, including the Serial Number. Then go to the Apple website that I mentioned above, and enter your Serial Number. If you have an iPhone 5 that is eligible for the program, you will get a message that "The iPhone 5 serial number you entered is eligible for this program. Please choose one of the service options below."
You don't need to rush to an Apple Store to get a new battery; the program lasts until two years after the first retail sale date of the iPhone 5 (which is September 21, 2014 if you bought it on the release date like I did) or March 1, 2015, whichever date provides more coverage. So in other words, you have at least six months from now, or perhaps longer if you bought your iPhone after March 1, 2013 (although Apple thinks that most affected units were sold before February of 2013). Of course, if your battery does seem to be exhibiting problems, you probably don't want to wait very long to get a replacement battery at an Apple Store or other authorized service provider.
I suspect that many iPhone 5 owners are nearing the end of a two-year contract, and thus plan to upgrade to the iPhone 6 when it is released next month (if you believe the rumors). I know that many folks pass on their used iPhones to family members or friends, and if you have an affected iPhone 5, then you might be able to get a fresh new battery for free before you pass along the device, which I'm sure that your family member or friend will appreciate.