`f(x) ` is the integral of `(2sinx-sin2x)/(x^3),x!=0.` Find `lim_(x->0)f^(prime)(x)[w h e r ef^(prime)(x)=(df)/(dx)]`

If f(x) is the integral of (2 sin x-sin 2 x)/(x^(3)), "where x" ne 0, "then find" lim_(x rarr 0) f'(x) .

If f(x) = int(sinx-sin2x)/(x^(3)) dx, where xne0 , then Limit_(xto0) f^(')(x) has the value

If f(x)=int((2sin x-sin2x)/(x^(3))dx);x!=0 then lim_(x rarr0)f'(x) is: (A) 0(B)oo(C)-1 (D) 1

Find lim_(x rarr0)f(x) and lim_(x rarr1)f(x), where f(x)=[(2x+3),x 0

A function f: R->R satisfies sinxcosy(f(2x+2y)-f(2x-2y)=cosxsiny(f(2x+2y)+f(2x-2y))dot If f^(prime)(0)=1/2,t h e n (a)f^(prime)(x)=f(x)=0 (b)4f^(x)+f(x)=0 (c)f^(x)+f(x)=0 (d)4f^(x)-f(x)=0

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these

If f(x)=|"cos"(x+x^2)"sin"(x+x^2)-"cos"(x+x^2)"sin"(x-x^2)"cos"(x-x^2)sin(x-x^2)sin2x0sin2x^2|,t h e n f(-2)=0 (b) f^(prime)(-1/2)=0 f^(prime)(-1)=-2 (d) f^(0)=4

Let the function f satisfies f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) foe all x and f(0)=3 The value of f(x).f^(prime)(-x)=f(-x).f^(prime)(x) for all x, is

Consider the function f(x)=1/x^(2) for x gt 0 . To find lim_(x to 0) f(x) .