Now the last I heard was that if someone could travel at c then time will have stopped for them. Yet seeing the above equation if v = c then you end up with deltat' = deltat/0. That to me doesn't say that time is stopped. That to me says that deltat' is either undefined or doesn't exist.
I believe that the deltat' is the change in time seen from an inertial reference frame moving with zero velocity, with respect to some moving reference frame with velocity v (I say zero velocity because if it is moving, but just not as fast, then the difference between the two speeds of the two different frames can be considered to be the speed of the first frame with respect to a non moving second frame, so the speed of the first frame is relative to the stationary one).
Having said that, if you were traveling in space quite fast with a velocity v and you looked back at a clock here on Earth, the successive ticks on the clock would appear to happen more quickly than the ticks on the clock on the wall inside your space vehicle. So with respect to Earth you are moving more slowly through time then people here on Earth and hence is the reason you can "travel into the future" if moving fast enough; because while you only may have seen 24 hours pass by aboard you ship, the time here on Earth that may have passed by could be 30 hours, so when you come home you traveled 6 hours "into the future".
You are not actually traveling into the future it is simply that the rate of time aboard your spaceship is "slower" than that on Earth, or time is moving more quickly on Earth with respect to time flow inside your ship. So when you get back to Earth it seems to everyone around you that you have been able to travel into the future as according to your clock you were only gone for say 24 hours while on Earth it appeared that you were gone for, say, 24 days.
So when viewed from your fast moving craft, the time on Earth appears to be moving at a faster rate than time aboard your ship. The faster you move the faster the ticks on the clock on Earth move as seen aboard your ship. Now if you take the limit of v approaching c what you find is that it appears that time on Earth compared to the flow of time aboard your ship has is moving at almost an infinite rate.
Your speed is so great that the successive ticks of a clock aboard your ship when viewed from Earth seem to take longer and longer between ticks. Again in the limit of v approaching c the next tick of the clock aboard your ship when viewed from Earth essentially never occurs; if you were actually capable of reaching the speed of c the infinite answer you get from the formula:
deltat' = deltat/sqrt(1 - (v/c)^2)
implies that the next tick of the clock aboard your ship when viewed from Earth is not just very, very far away from occurring, but will never occur. There will be no next tick for a clock aboard your ship when viewed from Earth as a change in time no longer exists on your ship when viewed from Earth.
So your ship's clock when viewed from Earth if you were somehow able to obtain the speed of light with your ship, would appear to have simply stopped moving all together. Therefore with respect to any outside observer time for you inside your ship has come to a halt. The only way to mathematically convey that there will never be another tick seen occurring on the clock inside your ship when viewed from Earth is to say the time between ticks on your clock aboard your ship
when viewed from Earth is infinite. That is there is no place on the time line in the future that, when viewed from Earth, the next tick of your clock will occur.
Saying there is an infinite amount of time between ticks on the clock aboard your ship when viewed from Earth is simply stating that the flow of time aboard your ship when viewed from Earth has come to a stop. The next tick of your clock occurs (again and like always when viewed from Earth) not just a very, very long time from now, but it occurs at a time that is infinitely far away from the current time. Since infinity does not exist on a number line, but rather is a concept, then it conveys that the next tick must never occur no matter how long you wait to see it change, it won't because the time between ticks on your clock when viewed from Earth have been spread infinitely far apart; which is the same as saying there will be no next tick on the clock aboard your ship ever when viewed from Earth.
I know I keep using Earth as the reference frame to compare the flow of time on your ship to, I'm doing so because it is convenient and I think it is easier to understand these concepts using Earth as a "stationary" inertial reference frame and a space ship moving with some velocity v with respect to Earth as the moving reference frame.
So yes, the mathematical formula will yield an infinite value if your ship is capable of reaching a speed of c, so it no longer gives you the time that needs to pass before another tick on your clock when viewed from Earth will occur; but rather by yielding an infinite result it is telling you that there will never be a time change aboard your ship when viewed from Earth; or put another way the amount of time that needs to pass in order for someone here on Earth to see your clock advance by one tick is infinitely long - restated - it will never occur because it is not just very, very far away on the time line, but it doesn't even exist on the time line, so there can never be any change in time aboard your ship when viewed from Earth.
Of course there is a problem even when just "thinking" about traveling at speed c. If time aboard your ship when viewed from outside your ship from Earth, Mars, Venus, wherever seems to have come to a stop aboard your ship when viewed from any of these places, the opposite is also true. That is time on Earth when viewed from your ship seems to be moving, again not just very,very quickly, but rather it seems to be passing at an infinite rate. So if you were able to obtain light speed with your ship, while all time aboard your ship when viewed from Earth appears to be stopped, to you all time viewed on Earth from your ship seems to pass by in an instant. So you get to see all future events at the same instant of time; and if time does come to an end someday then you'll be there to see it end. If time goes on forever, as is currently assumed, then you will see all future events happen all at once; talk about sensory overload, whew! Never mind the fact that if you could reach light speed your mass would become infinite in amount which would require an infinite amount of energy to keep you moving at speed c. I don't think there's an infinite amount of energy or mass within our universe, so basically one can never obtain the speed of light unless one becomes a photon.
Another effect of being able to obtain the speed of c is that because time and space are interwoven concepts; that is they coexist together. When you look out and see time passing at an infinite rate, space is also infinitely condensed into a point; so you are everywhere at the same time watching time pass by at an infinite rate. These are of course things that can't ever happen to us, but they are a direct result of the mathematics when your speed v is replaced with speed c. Of course even if you could reach the speed of c with your ship, the clocks aboard your ship when viewed by you (also aboard the ship) are moving at the normal rate we are used to seeing time move at, even though when viewed from Earth it seems that time has come to a stop for you and your ship, aboard it time seems to be moving normally; that's why it's called relativity. The time seen passing for something moving at a different speed than you are appears to be moving either more quickly or more slowly relative
to your reference frame; and like wise the flow of your time seen from the other differently speeding object is also different than the flow of time with respect to that same object.
So, for you, no matter how fast you are moving, time seems to flow at a constant rate when you look at your watch. If you look at other objects whose speeds are different than yours, the rate at which time passes for these other objects when viewed by you is different than the flow of time you see when you look down at your watch. So the flow of time for things that are moving with a different speed with respect to you, when viewed by you, is relative to the difference in the speed of these other things and your speed.
I hope that helps. I do understand what you are getting at in that the function yields infinity for a speed of c for v and not an actual number. It just means that there is no defined length of time that exists such that the clock moving at speed c will change in time, i.e. tick even once. Which is the same as saying if there is an infinite value for an amount of time that is needed before the clock that is moving at speed c when viewed to tick even once, then it will simply never tick again as long as it is moving at speed c because the next tick is infinitely far away, or put another way simply doesn't exist; therefore if the next moment in time can never be reached aboard your ship because it doesn't exist when viewed from something moving slower than c, than time must be stopped aboard your ship when viewed from something not moving at speed c; even though you on your ship when you look at your clock it seems to be moving normally it appears to have stopped to everything viewing it that is not moving at the same speed you are.