What's that limit, where we start to see infrared galaxies? All the nearby galaxies we see come in visible range, many are visible from ground based telescopes. And the farthest galaxy that hubble discovered recently, 13.4 billion light years away is in infra red.
Was this Galaxy ever in visible part of spectrum? 
How long will it take galaxies like andromeda or whirlpool to go infrared so that they won't be visible to us if we are not using in infraRed lens?
What's the specific limit where the light makes transition from visible to infrared part of spectrum?
Galaxies give off a range of frequencies of light: Visible, but also infrared, microwave, and radiowaves. at longer wavelengths, ultraviolet, Xrays and gamma rays at short wavelengths. But stars give off quite a lot of their radiation in visible light, so nearby galaxies are quite bright in visible light and rather less bright in Ultraviolet.

Very distant galaxies are moving away from us due to the general expansion of the galaxy. This means that the light is stretched. Visible light gets stretched to infra-red, ultraviolet gets stretched to visible light. The distant galaxy is very very far away, and so very dim. It is a little bit brighter in infrared, although there is also a little visible light (and other wavelengths too) 

Galaxies never go completely invisible in visible light, even if they are red-shifted, because there is always ultraviolet light to red-shift to visible.

The specific limit is often taken to be 700nm as the boundary between red and infrared. Although this is fuzzy because some people can see a little further than others.