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 continuous at x=0 If lim_(xto0) (2f(x)-3af(2x)+bf(8x))/(sin^(2)x) exists and f(0)ne0,f'(0)ne0 , then the value of 3a//b is________.

Let f''(x) be continuous at x = 0 and f"(0) = 4 then value of lim_(x->0)(2f(x)-3f(2x)+f(4x))/(x^2)

Let f(x) be a twice-differentiable function and f"(0)=2. The evaluate: ("lim")_(xvec0)(2f(x)-3f(2x)+f(4x))/(x^2)

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 a function satisfies the relation f(x) f''(x)-f(x)f'(x)=(f'(x))^(2) AA x in R and f(0)=f'(0)=1, then The value of lim_(x to -oo) f(x) is

Let f(x) be a function such that ("lim")_(xvec0)(f(x))/x=1 and ("lim")_(xvec0)(x(1+acosx)-bsinx)/((f(x))^3)=1 , then b-3a is equal to

If f (x) is a thrice differentiable function such that lim _(xto0)(f (4x) -3 f(3x) +3f (2x) -f (x))/(x ^(3))=12 then the vlaue of f '(0) equais to :

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

If f (x) is a thrice differentiable function such that lim _(xto0)(f (4x) -3 f(3x) +3f (2x) -f (x))/(x ^(3))=12 then the vlaue of f '''(0) equais to :

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