The video below shows how the British police handled a high-risk incident - a report of a mentally ill man wandering the streets in his pyjamas, brandishing a handgun. The suspect is believed to have returned to his flat by the time they arrive on scene, so the officers literally knock on his door and politely ask if he has a gun.

With the right training, American officers can behave in exactly the same manner.

https://youtu.be/LKVyu1sodOU?t=3m35s