There are two definitions of 234 . Definition 1 gives
(23)4 which is 212 and definition 2 gives
2(34) which is 281.
Similarly the values of (Ö2Ö2)Ö2
and Ö2(Ö2Ö2) are not equal. The
first of these is f(f(Ö2)) where
f(x) = xÖ2 ; the second of these is g(g(Ö2))
where g(x) = (Ö2)x.
To see what happen if you iterate the functions many times you should now
experiment, using your
calculator or computer, by iterating both f and g in each case
starting with the value Ö2.
Using these two definitions, we think of
(where the powers of root 2 go on for ever) as the limit as n to infinity of
the sequence
where, according to the first definition, xn+1 = f(xn), or equivalently,
and, according to the second definition, xn+1 = g(xn), or
equivalently,
In both cases, if the limit exists, you will find it by putting
xn+1 = xn = x.