Let `f(0)=0` and `f'(0)=1`. For a positive integral `k` show that `lim_(x->o) 1/x(f(x)+f(x/2)+....f(x/k))=1+1/2+1/3+...+1/k`

Let f(0)=0 and f'(0)=1. For a positive integral k show that lim_(x rarr o)(1)/(x)(f(x)+f((x)/(2))+....f((x)/(k)))=1+(1)/(2)+(1)/(3)+...+(1)/(k)

Let f(x)=x sin x - (1-cosx) . The smallest positive integer k such that lim_(x rarr0)(f(x))/(x^(k))!=0 is

Let f(x) be defined for all x in R such that lim_(xrarr0) [f(x)+log(1-(1)/(e^(f(x))))-log(f(x))]=0 . Then f(0) is

If f'(x)=k,k!=0, then the value of lim_(x rarr0)(2f(x)-3f(2x)+f(4x))/(x^(2)) is

If f(x) be a differentiable function for all positive numbers such that f(x.y)=f(x)+f(y) and f(e)=1 then lim_(x rarr0)(f(x+1))/(2x)

If f(x)=det[[sin x,cos x,tan xx^(3),x^(2),x2x,1,1]] then show that lim_(x rarr0)(f(x))/(x^(2))=1

Let f:(0,1) in (0,1) be a differenttiable function such that f(x)ne 0 for all x in (0,1) and f((1)/(2))=(sqrt(3))/(2) . Suppose for all x, lim_(x to x)(int_(0)^(1) sqrt(1(f(s))^(2))dxint_(0)^(x) sqrt(1(f(s))^(2))ds)/(f(t)-f(x))=f(x) Then, the value of f((1)/(4)) belongs to

Let f(x) be a polynomial of least degree such that lim_(x rarr0)(1+(x^(4)+f(x^(2)))/(x^(3)))^(1/x)=e^(3), then write the value of f(3)

Let f:RtoR be such that f(1)=3 and f'(1)=6 . Then lim_(xto0)[(f(1+x))/(f(1))]^(1//x) equals