6.If `f(0) = 0= g(0)` and `f'(0) =6 = g'(0)` then `lim_(x->0) f(x)/g(x)`

Let f(x+y) = f(x) f(y) and f(x) = 1 + x g(x) G(x) where lim_(x->0) g(x) =a and lim_(x->o) G(x) = b. Then f'(x) is

Let x=f(t) and y=g(t), where x and y are twice differentiable function. If f'(0)= g'(0) =f''(0) = 2. g''(0) = 6, then the value of ((d^(2)y)/(dx^(2)))_(t=0) is equal to

If f(x)=sgn(x)" and "g(x)=x^(3) ,then prove that lim_(xto0) f(x).g(x) exists though lim_(xto0) f(x) does not exist.

If f(x), g(x) be twice differentiable functions on [0,2] satisfying f''(x) = g''(x) , f'(1) = 2g'(1) = 4 and f(2) = 3 g(2) = 9 , then f(x)-g(x) at x = 4 equals (A) 0 (B) 10 (C) 8 (D) 2

If f(x), g(x) be twice differentiable functions on [0,2] satisfying f''(x) = g''(x) , f'(1) = 2g'(1) = 4 and f(2) = 3 g(2) = 9 , then f(x)-g(x) at x = 4 equals (A) 0 (B) 10 (C) 8 (D) 2

Let fandg be real valued functions defined on interval (-1,1) such that g''(x) is constinous, g(0)=0 , g'(0)=0,g''(0)=0andf(x)=g(x)sinx . Statement I lim_(xrarr0)(g(x)cotx-g(0)cosecx)=f''(0) Statement II f'(0)=g'(0)

Let f : R ->(0,oo) and g : R -> R be twice differentiable functions such that f" and g" are continuous functions on R. suppose f^(prime)(2)=g(2)=0,f^"(2)!=0 and g'(2)!=0 , If lim_(x->2) (f(x)g(x))/(f'(x)g'(x))=1 then

Let f be a differentiable function satisfying f(xy)=f(x).f(y).AA x gt 0, y gt 0 and f(1+x)=1+x{1+g(x)} , where lim_(x to 0)g(x)=0 then int (f(x))/(f'(x))dx is equal to

If f(x) and g(x) are differentiable function for 0 le x le 23 such that f(0) =2, g(0) =0 ,f(23) =22 g (23) =10. Then show that f'(x)=2g'(x) for at least one x in the interval (0,23)

if int_(0)^f(x) 4x^(3)dx=g(x)(x-2) if f(2)=6 and f'(2)=(1)/(48) then find lim_(x to 2) g(x)