So again - no, telescopes do not have any limited distance range. The only limit is the size of the observable Universe.
In practice though, a
specific telescope has limits. But it we are talking about a telescope with arbitrary parameters (light collecting area, exposure time), then no, there's no real limit.
@Justatruthseeker started out with these essentially correct observations (assuming the sensitivity of the sensor is about he same as the eye
Assuming that a star is so far away that it is barely visible to the naked eye, we know that the Hubble telescope can make the star appear 127,551 times brighter. Does this mean that the Hubble telescope enables an observer to see the star if it were 127,551 times farther away? The answer is no. The Inverse Square Law says that the light that we receive from a star is inversely proportional to the square of its distance. According to this law, at that distance, the light of the star becomes 127,551^2 or 16,269,262,700 times dimmer, far too dim for us to see with the telescope.
This raises the question: What is the maximum distance an object can be seen through the Hubble telescope? The answer is 357.14 times the distance that the naked eye can see. The reason is that an object 357.14 times farther away, its light becomes 127,551 times dimmer. Since the Hubble telescope can make a star appear 127,551 times brighter, then looking through the telescope the star would be barely visible.
Not unreasonable observations. However he then goes on to totally discount the effect of long exposures.
We can frame the problem using JAT's scenario of a star (or galaxy) that is barely visible to the naked eye. How much further could the Hubble see this same star from?
As the light diminished by the inverse square of the distances, then
any increase we achieve in acquiring light will only increase that distance by the square root of the increase in light. So as JAT correctly explains, an increase in light receiving area of 127,551x will push the distance to the square root of that, or 357x.
The real light collecting power comes in the form of long exposures. Increase in the length of exposure is directly proportional to increase light collected by the sensor. However,
any increase we achieve in acquiring light will only increase that distance by the square root of the increase in light. So if we increase the exposure by a factor of 100, we will only push the distance 10x.
However, the key to the Hubble is that it can do
really long exposures, many hours, hundreds of thousands of second, or
millions of times longer than the eye exposure equivalent (1/10th of a second, being conservative). So we can gather millions of times more light. But
any increase we achieve in acquiring light will only increase that distance by the square root of the increase in light, so really it's "only" thousands of times more distance
So to repeat my earlier calculation to fit this more long-winded but hopefully more understandable explanation, if the Hubble has an exposure of 350,000 seconds (as per wikipedia) then that's an increase in light of 3,500,000, and of course
any increase we achieve in acquiring light will only increase that distance by the square root of the increase in light, hence it's only 1870x more distance.
So we've got 357x the distance for the bigger light collecting area, and 1870x the distance for the longer exposure, so that's 667,000 times the distance with both combined.
So (in our simplified example) if you can see the Andromeda galaxy with the naked eye at 2.5 million light years, then the Hubble can photograph it at 1.6 Trillion light years. Or it could see an object that's 1% the brightness of Andromeda at 16 billion light years.
[Note: Yes I'm repeating myself a bit here, I'm just interested in how best to explain things, so I thought I'd give it another go]