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How do I look up the battery charge cycles on an iPhone 4s? Is there a way to do it right on the phone? Or, do I have to connect it to my computer and then look it up somehow with iTunes, Console, or something else?

You will need to use a battery utility, such as coconutBattery (for Mac) or 3uTools (for Windows). It will tell you what the health of the battery is. Anything less than 70% of design capacity will require replacement.