Q: Using iOS 7 on iPad(3 & 4) and connecting to Apple TV

Try this  - Reset the iPad by holding down on the Sleep and Home buttons at the same time for about 10-15 seconds until the Apple Logo appears - ignore the red slider - let go of the buttons. (This is equivalent to rebooting your computer.)

http://support.apple.com/kb/ht5209

http://support.apple.com/kb/ht4437

http://support.apple.com/kb/TS4215

 Cheers, Tom

Not so fast. Yes, after reseting the iPad mirroring works, until about the 4th connection to airplay and then the problem comes right back. This is definitely an iOS 7 issue with the iPad 3rd generation for sure. I have 3 of them in my classroom and they all have the same problem. iOS 6 had no issues at all with Apple TV. Apple needs to address this problem.

I agree. We are having this problem at our school, where many of the teachers teach with iPads and Apple TV's solely for the purpose of wireless connection from the iPad to the projector. And ever since the iOS 7 upgrade, we are having exactly the issue you describe: every few times you try to connect, you get a black screen on the projector (Apple TV) and the iPad suddenly gets VERY slow. Resetting is the only solution--which takes time. This needs to be addressed by Apple ASAP. - JR

iOS 7.1 seems to have fixed the airplay issues with Apple TV. After updating my iPad 3 to iOS 7.1 and Apple TV to 6.1 the  iPad 3 mirrors without any issues.