The value of `underset(xtooo)lim((2^(x^(n)))e^((1)/(x))-(3^(x^(n)))e^((1)/(x)))/(x^(n))` (where `n in N`) is

The value of lim_(xtooo) ((2^(x^(n)))e^((1)/(x))-(3^(x^(n)))e^((1)/(x)))/(x^(n)) (where n in N ) is

underset(n to oo)lim(((n+1)(n+2)...3n)/(n^(2n)))^((1)/(n)) is equal to

The value of lim_(ntooo)(e^(n))/((1+(1)/(n))^(n^(2))) is

The value of lim_(ntooo) [(1)/(n)+(e^(1//n))/(n)+(e^(2//n))/(n)+...+(e^((n-1)//n))/(n)] is

Show : underset(xrarr2)"lim"((1+x)^(n)-3^(n))/(x-2)=n.3^(n-1)

Statement -I : underset(xrarroo)"lim"(1^(2)/(x^(3))+(2^(2))/(x^(3))+3^(2)/(x^(3))+......+x^(2)/(x^(3)))= underset(xrarroo)"lim"(1^(2))/(x^(3))+underset(xrarroo)"lim"(2^(2))/(x^(3))+....+underset(xrarroo)"lim"(x^(2))/(x^(3))=0 Statement - II : underset(xrarra)"lim"{f_(1)(x)+f_(2)(x)+....+f_(n)(x)}=underset(xrarra)"lim"f_(1)(x)+underset(xrarra)"lim"f_(2)(x)+...+underset(xrarra)"lim"f_(n)(x)"

x^(3m)+x^(3n-1)+x^(3r-2) where m,n,r in N is divisible by

lim_(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2)), n in N , equals

If the value of the definite integral int_0^1(sin^(-1)sqrt(x))/(x^2-x+1)dx is (pi^2)/(sqrt(n)) (where n in N), then the value of n/(27) is

lim_(xto0) ((2^(m)+x)^(1//m)-(2^(n)+x)^(1//n))/(x) is equal to