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Record-Breaking Galaxy Found In Deep Hubble Image
The Bad Astronomer writes "Astronomers using Hubble Space Telescope have found a galaxy at the very edge of the Universe: the light from this far-flung object has been traveling a whopping 13.1 billion years to get here! The galaxy appears as a non-descript dot in the infrared Hubble Ultra Deep Field taken using the Wide Field Camera 3, but a spectrum taken using a ground-based telescope confirms that we're seeing this object as it was a mere 600 million years after the Big Bang itself."
No, there is no way to know for sure if it still exists, but I think most don't "live" that long and it has probably faded out or "evolved" into something different.
If you could reason with religious people, there would be no religious people
I think there's a maximum length after which a galaxy cannot exist; diminishing element returns from supernovae. Unfortunately I'm not sure how long it is, but it's much longer than 13 billion years; individual red dwarves can last for hundreds of billions of years. As for merger with other galaxies or destruction by a supermassive black hole though, its anyone's guess.
"People don't want to learn linux" hasn't been a valid excuse since '03.
according to relativity, if we see it it exists.
This question is not well phrased. There is no universal "now" in relativity. You probably mean something like "in our reference frame does this galaxy exist somewhere now", and then the answer is that we can't tell. If you'll choose some other reference frame, you'll get different points to correspond to our "now". So abandon the notion of "still exist", it exists "now" in the most meaningful way, the point we see when we look there...