Evaluate
`lim_(xrarrc) f(x)` where `f(x) = a_nx^n+a_(n-1)x^(n-1)+a_(n-2)x^(n-2)+.....+a_2x^2+a_1x^1+a_0`

Differentiate |x|+a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-1)+...+a_(n-1)x+a_(n)

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

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

(1+x)^(n)=a_(0)+a_(1)x+a_(2)*x^(2)+......+a_(n)x^(n) then prove that

Differentiate the following with respect of x:a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)++a_(n-1)x+a_(n)

Let nge3 be an integer. For a permutaion sigma=(a_(1),a_(2),.....,a_(n)) of (1,2,.....,n) we let f_(sigma)(x)=a_(n)X^(n-1)+a_(n-1)X^(x-2)+....+a_(2)x+a_(1) . Let S_(sigma) be the sum of the roots of f_(sigma)(x)=0 and let S denote the sum over all permutations sigma of (1,2,.....,n) of the numbers S_(sigma) . Then-

a_(0)f^(n)(x)+a_(1)f^(n-1)(x)*g(x)+a_(2)f^(n-2)g^(2)(x)+......+a_(n)g^(n)(x)>=0

If n ge 2 , and (1 + x + x^(2))^(n) = a_(0) + a_(1)x + a_(2)x^(2) + .. . +a_(2n) x^(2n) , then a_(0) - 2a_(1)+ 3a_(2) + ... + (2n + 1)a_(n) = ______

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

Let f(x)=a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+......+a_(n),(a_(0)!=0) if a_(0)+a_(1)+a+_(2)+......+a_(n)=0 then the root of f(x) is