tiling near carpet

My tiling ability is limited. I have tiled a backsplash in a kitchen before, but that is about it. I am about to undertake another tiling project, but have a question. I am going to be tiling a hearth. The hearth buts up against carpet and there is no metal edge between the two. So, my question is, after I tear out the old tile and place down any backing I need, how do I re-tile near the carpet and make sure the carpet is still securly fastened to its tack strip? Basically, what do I do when the tile and the carpet meet to make it look like a smooth transition (like it currently does). Thanks in advance for any help you can give.

What is under the existing tile (if anything)?
Why can't the carpet stay as-is? Don't disturb it and relay up to it again.