If `lim_( n ->alpha) x^(1/n) = 1 ` then find the value of `alpha`

If lim_(n rarr oo)(1)/((sin^(-1)x)^(n)+1)=1, then find the value of x.

If lim_(x to alpha) (alpha^x - x^alpha)/(x^alpha - alpha^alpha) = -1 then what is the value of alpha ?

If alpha and beta are roots of the equation x^(2)-3x+1=0 and a_(n)=alpha^(n)+beta^(n)-1 then find the value of (a_(5)-a_(1))/(a_(3)-a_(1))

If u_(n) = cos^(n) alpha + sin^(n) alpha , then the value of 2 u_(6) - 3 u_(4) +1 is :

If f(alpha)=f(beta) and n in N , then the value of int_alpha^beta (g(f(x)))^n g\'(f(x))*f\'(x)dx= (A) 1 (B) 0 (C) (beta^(n+1)-alpha^(n+1))/(n+1) (D) none of these

If 1,alpha,alpha^(2),alpha^(3),......,alpha^(n-1) are nn^(th) roots of unity,then find the value of (2011-alpha)(2011-alpha^(2))....(2011-alpha^(n-1))