Black holes are indeed at the centers of every large galaxy like our own Milky Way, and our Milky Way's black hole is a good example of 'normal' black hole behavior, at least for galaxies living near us.
But as a first note, we have to clear up a few misconceptions. Black holes don't inhale material from their surroundings; they're extreme gravitational objects, but the force of gravity is their only mechanism for drawing material inwards towards them. A black hole won't inhale planets that orbit around it any more than a star will. The supergiant black holes at the centers of galaxies, by the same token, won't inhale stars and gas at a faster rate than any other object of the same mass.
The other facet to your question is on the 'death' of a black hole, which is an interesting concept. What would it mean for a black hole to die? Will the gravitational distortion associated with a black hole disappear if it fails to bring in new material? To that question we can give a definitive answer. Black holes are very stable objects, and will remain static and unchanging if there is no new material which interacts with the black hole. If you sit a black hole in the void of space, out on its lonesome, it will simply sit there. It won't be able to grow any more massive (and by proxy it won't grow physically larger either), but large black holes won't disappear, either. It's very difficult to destroy mass in our universe, and as one way to think about black holes is as a very dense concentration of matter, the destruction of a black hole is no simple thing.
If, on the other hand, we expand a definition of the black hole's death to include a black hole going from a luminous, bright object in the astrophysical sky to something that is invisible by almost all methods of looking at it, then yes; the lack of material around a black hole can cause a black hole to die. This is a dying down, rather than a fading from existence in the Universe, but if the black hole has been trying to grow its mass, it glows brightly, and if that material is suddenly consumed, that glow can be shut down rapidly, the death of the black hole as a light source.
Going back to our own Milky Way's black hole - currently it has nothing to feast on. Its existence is under no threat, and the black hole, as a large astrophysical object, will remain in place and unperturbed until some star or gas cloud drifts too near. At that point, the black hole will shred the star or the looser cloud of gas, heating it until it glows brightly to a number of telescopes based here on Earth. (Not to worry; anything that happens in our Galaxy's black hole is sufficiently far away that it should pose no danger to us.) Once that gas is consumed? The black hole will subside into darkness once more, perhaps a little larger after its meal; dark but not lost.
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