`lim_(x->0)1/(x^(k+1))int_0^x(x^k+x^(k-1)y+x^(k-2)y^2+.....+y^k)f(y)dy=`

Y = sum_ (k = 0) ^ (5) x ^ (k) a ^ (k-1)

Let x gt, 0, y gt 0, z gt 0 are respectively the 2^(nd), 3^(rd), 4^(th) terms of a G.P. and Delta = |(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2))|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

If the normal to the curve y=f(x) at x=0 be given by the equation 3x-y+3=0 , then the value of lim_(x rarr0)x^(2){f(x^(2))-5f(4x^(2))+4f(7x^(2))}^(-1) is (-1)/(k) then k =

det [[x ^ (k), x ^ (k + 2), x ^ (k + 3) y ^ (k), y ^ (k + 2), y ^ (k + 3) z ^ (k) , z ^ (k + 2), z ^ (k + 3)]] = (xy) (yz) (zx) {(1) / (x) + (1) / (y) + (1) / ( from)}

Evaluate lim_(x to 0) (k^([x]) - 1- |x| In K)/(x^(2)) , k gt 0

Evaluate lim_(x to 0) (k^([x]) - 1- |x| In K)/(x^(2)) , k gt 0

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then