Let `f ''(x)` be continuous at `x=0` If `("lim")_(xvec0)(2f(x)-3a(f(2x)+bf(8x))/(sin^2x)` exists and `f(0)!=0,f^(prime)(0)!=0,` then the value of `(3a)/b` is____

Let f(x) be a function defined on (-a ,a) with a > 0. Assume that f(x) is continuous at x=0a n d(lim)_(xvec0)(f(x)-f(k x))/x=alpha,w h e r ek in (0,1) then f^(prime)(0^+)=0 b. f^(prime)(0^-)=alpha/(1-k) c. f(x) is differentiable at x=0 d. f(x) is non-differentiable at x=0

If f(x)=inte^(x)(tan^(-1)x+(2x)/((1+x^(2))^(2)))dx,f(0)=0 then the value of f(1) is

f(x)={x ,xlt=0 1,x=0,then find ("lim")_(xvec0) f(x) if exists x^2,x >0

If f(x) = int(5x^(8)+7x^(6))/((x^(2)+1+2x^(7))^(2))dx, (x ge 0) , and f(0) = 0, then the value of f(1) is

If f((x+2y)/3)=(f(x)+2f(y))/3AAx ,y in Ra n df^(prime)(0)=1,f(0)=2, then find f(x)dot

Let f(x) be a twice-differentiable function and f''(0)=2. Then evaluate lim_(xto0) (2f(x)-3f(2x)+f(4x))/(x^(2)).

Let f(x) be a polynomial satisfying lim_(xtooo) (x^(2)f(x))/(2x^(5)+3)=6" and "f(1)=3,f(3)=7" and "f(5)=11. Then The value of f(0) is

If f(x) is twice differentiable and f^('')(0) = 3 , then lim_(x rarr 0) (2f(x)-3f(2x)+f(4x))/x^(2) is

If y=f(x^3),z=g(x^5),f^(prime)(x)=tanx ,a n dg^(prime)(x)=secx , then find the value of (lim)_(xvec0)(((dy)/(dz)))/x

Suppose f(x)=e^(a x)+e^(b x) , where a!=b , and that f prime prime(x)-2f^(prime)(x)-15f(x)=0 for all xdot Then the value of (|a b|)/3 is ___