Evaluate : `lim_(x->0)((1-cos(a_1x)*cos(a_2x)*cos(a_3x)).........cos(a_n x))/(x^2).` where `a_1,a_2,a_3...........a_n in R.`

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_3+x)+ ........+ 1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to (a) pi (b) 2pi (c) 3pi (d) pi/2

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_3+x)+ ........+ 1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to (a) pi (b) 2pi (c) 3pi (d) pi/2

let f(x)=a_0+a_1x^2+a_2x^4+............a_nx^(2n) where 0< a_0 < a_1 < a_3 ............< a_n then f(x) has

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_3+x)++1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to pi (b) 2pi (c) 3pi (d) pi/2

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_1+x)++1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to

If (1-px)^-1/((1-qx))=a_0+a_1x+a_2x^2+a_3x^3+....... then a_n=

Let a_1,a_2,.....,a_n be fixed real numbers and define a function f(x) = (x-a_1) (x-a_2).....(x-a_n) . What is (lim)_(x->a_1)f(x) ? For some a!=a_1,a_2,.....,a_n , compute (lim)_(x->a)f(x)

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_1+x)++1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to pi (b) 2pi (c) 3pi (d) pi/2

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_1+x)++1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to pi (b) 2pi (c) 3pi (d) pi/2