`underset(n→∞)limx^(1/n) = 1,x>0`

Prove that underset( n rarr oo)lim x_(n)=1, if x_(n)=(3^(n)+1)/3^(n) .

(i) Let h (x) = underset(x to oo)lim(x^(2n) f(x) + g(x))/(1+x^(2n)) , find h(x) in terms of f(fx) and g(x) (ii) without using L Hospital rule or series expansion for e^(x) evaluate underset(x to 0) lim (e^(x) -1-x)/x^(2) (iii) underset(n to oo) lim [ e^(1/n)/n^(2) + 2 ((e^(1/n))^(2))/n^(2) + 3. ((e^(1/n))^(3))/n^(2)+.......+ n((e^(1/n))^(n))/n^(2)] (iv) underset(x to 0)lim[ (a sin x)/x ] + [ (b tan x)/x] Where a,b are inegers and [] denotes integral part. (v) underset(x to a)lim (sinx/sina)^(1/(x-a))

Taking advantage of the theorem on passing to the limit in inequalities, prove that underset(n to oo)lim x_(n)=1 if x_(n)=2n(sqrt(n^(2)+1)-n)