Question

an = square root of 3, an+1 = square root of 3+ an
show by induction or otherwise thatt an is inreasing and bounded above by 3

Answer 1

assuming you meant
a_1 = √3
a_n+1 = √(3+a_n)
using the ratio test,
a_n+1 / a_n = √(3+an)/an
either an is bounded -- you're done
or an is unbounded. In that case, you have in the limit,
√(3+an)/an → √an / an = 1/√an = 0
so an is not unbounded

As for the value of the limit, see item #3 in this helpful article
(which you could have found had you tried)

sites.math.washington.edu/~morrow/336_11/papers/brian.pdf