My iPhone keeps dying with 20% battery. What should I do?

I have an iPhone 5 and recently it has been dying with like 20% battery or it will start dropping insanely fast when it get below 40%
Today it died with 40%
It has died before because it got too cold but that's not what it is this time. Or it shouldn't be be.
I kill my apps frequently and turn down my brightness, but is there anything I can do to fix this?
Also I have insurance through verizon, do you thing they would replace it for this? They used to replace my old blackberry for anything.

Try turning it back on when it dies, and do this a few times. Then put the phone on charge for at least 10 hours to reset the battery's half life. If that doesn't work, then there's a fault with the battery and it should be replaced. If you have not had the phone for more than 12 months it should be covered by the warranty.

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