Exapnd the expression as a Taylor series, noting that the first term is √(x²+a) so that as y-> infinity using only the first term in the expansion will be increasingly accurate. The expansion gives:
√y + 1/2 (√(y+√y))/√y) - √y =
1/2*√(1 + 1/√y)
In the limit, 1/√y approaches zero so the outer √ term approaches one and the overal limit is 1/2.
a different approach to the same solution
