Are you sure it's not
and ?
If so, this is a relatively easy limit to evaluate...
It should be clear that the sequence goes .
So you want to see if can be evaluated (has a limit...). Call this limit .
If then
or
or .
It should be clear that is an extraneous solution which came from the original squaring of the equation, so the limit is .
The difference equation can be written as...
(1)
The function is illustrated here...
There is only one 'attractive fixed point' in but that's only a necessary condition of convergence. In this case however is [red line...] so that any will generate a sequence monotonically convergent at ...
Kind regards