Calculate the limit: lim [(n^2 + 1)/(n^2 +2)]^[(n+1)/(n+2)], n tends to infinity value.

Lim{(n^2+1)/(n^2+2) }^{(n+1)/(n+2)}

=Lim {(1-1/(n^2+2))^(n^2+2)} ^{( n+1)/[n^2+2)(n+2)]}.

=Lim{(1-1/N)^N }^ {n+1/[N(n+2)]},Where,  n^2 +2 =N ->   infinite as n-> infnite.

=(1/e)^0 = 1 , as  the expression in the power is

( n+1)/N(n+2)} = Lim (n+1)/[(n^2+2)(n+2)] =  [(1+1/n)/( n^2+n+2+2/n] -> 1/infinitre ->0 as n approache infinitely high.

Thus the Lim [(n^2+1)/(n^2+2]^[(n+1)/(n+2)] = 1