It doesn't make sense to ask for the limit of the numbers between 1 and 2.
That collection certainly isn't a sequence, in the ordinary sense. That collection isn't a function either.
So, it doesn't make sense to ask for its limit as if it were a sequence, or as if it were a function.
Now, there are interesting things you can ask -- you can ask things like:
"What is the least upper bound of (1, 2)"?
"What is the greatest lower bound of (1, 2)"?
"What is the cardinality of (1, 2)"?
"What is the order type of (1, 2)"?
Or, you might write a function like
f(x) = (3/2) + (1/pi) arctan(x)
whose image is precisely the interval (1, 2), and then you could ask questions like:
"What is the limit of f(x) as x goes to -infinity?"