in the limit

puntmat posted an interesting idea (7th August 2014) involving various functions and exploring their limit as n gets very large indeed (tends to infinity)

these resources involve two of their functions plus another one

what fractions are shaded with a colour (out of the whole rectangle) at each stage?
what happens to this fraction as the shape (n) grows bigger and bigger?

all give rise to a common set (or subset) of fractions
with the same limit...
why?

the puntmat post helpfully included some animated gifs for two of these sequences: