" 11."lim_(x rarr oo)((a_(1)^(t)+a_(2)^(t)+a_(1)^(t)+...+u_(n)t)/(n))^(n)" where "a_(1),a_(2),a_(3),quad a_(n)>0

lim_(x rarr0)((a_(1)^(x)+a_(2)^(x)......+a_(n)^(x))/(n))^((1)/(x))=

lim_(x rarr oo){((a_(1))^((1)/(x))+(a_(2))^((1)/(x))+...+(a_(n))^((1)/(x)))/(n)}^(nx)

If a_(1),a_(2),a_(3),a_(4),,……, a_(n-1),a_(n) " are distinct non-zero real numbers such that " (a_(1)^(2) + a_(2)^(2) + a_(3)^(2) + …..+ a_(n-1)^(2))x^2 + 2 (a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) + ……+ a_(n-1) a_(n))x + (a_(2)^(2) +a_(3)^(2) + a_(4)^(2) +......+ a_(n)^(2)) le 0 " then " a_(1), a_(2), a_(3) ,....., a_(n-1), a_(n) are in

If a_(1),a_(2),a_(3),a_(4),,……, a_(n-1),a_(n) " are distinct non-zero real numbers such that " (a_(1)^(2) + a_(2)^(2) + a_(3)^(2) + …..+ a_(n-1)^(2))x^2 + 2 (a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) + ……+ a_(n-1) a_(n))x + (a_(2)^(2) +a_(3)^(2) + a_(4)^(2) +......+ a_(n)^(2)) le 0 " then " a_(1), a_(2), a_(3) ,....., a_(n-1), a_(n) are in

Evaluate : lim_(x rarr0)((1-cos(a_(1)x)*cos(a_(2)x)*cos(a_(3)x))......cos(a_(n)x))/(x^(2)) where a_(1),a_(2),a_(3).........a_(n)in R

Evaluate: lim_(n rarr oo)((a_(1)^((1)/(1))+a_(2)^((1)/(2))+...+a_(n)^((1)/(n)))/(n))^(nx)

lim_(x rarr oo)[((x+a_(1))(x+a_(2))dots.......(x+a_(n)))^((1)/(n))-x]

Let f(x) = a_(0)x^(n) + a_(1)x^(n-1) + a_(2) x^(n-2) + …. + a_(n-1)x + a_(n) , where a_(0), a_(1), a_(2),...., a_(n) are real numbers. If f(x) is divided by (ax - b), then the remainder is

Evaluate underset( n rarroo)("lim") ((a_(1) + 1)/( a_(1))) ((a_(2) +1)/(a_(2))) "......." ((a_(n) +1)/( a_(n))) where a_(1) 1 and a_(n) = n ( 1+ a_(n-1) ) AA n ge 2