Let `f` be differentiable at `x=a\ a n d\ l e t\ f(a)!=0.` Evaluate `(lim)_(n->oo){(f(a+1/n))/(f(a))}^ndot`

Let f be a positive differentiable function defined on (0,oo) and phi(x)=lim_(nrarroo) (f(x+(1)/(n))/f(x))^(n) . Then intlog_(e)(phi(x))dx=

Let f:R to R differerntiable at x=0 and satisfies f(0)=0 and f'(0)=1, then the value of lim_(x to 0) (1)/(x) sum_(x to 1)^(oo) (-1)^(n) f((x)/(n))is

let f(x)=lim_(n rarr oo)(x^(2n)-1)/(x^(2n)+1)

Let f(x) be differentiable on the interval (0,oo) such that f(1)=1 and lim_(t rarr x)(t^(2)f(x)-x^(2)f(t))/(t-x)=1 for each x>0 Then f(x)=

Let f : R toR " be a real function. The function "f" is double differentiable. If there exists "ninN" and "p inR" such that "lim_(x to oo)x^(n)f(x)=p" and there exists "lim_(x to oo)x^(n+1)f(x), "then" lim_(x to oo)x^(n+1)f'(x) is equal to

Let f : R toR " be a real function. The function "f" is double differentiable. If there exists "ninN" and "p inR" such that "lim_(xto oo)x^(n)f(x)=p" and there exists "lim_(x to oo)x^(n+1)f(x), "then" lim_(x to oo) x^(n+2)f''(x) is equal to

Let f:(0, oo)rarr(0, oo) be a differentiable function such that f(1)=e and lim_(trarrx)(t^(2)f^(2)(x)-x^(2)f^(2)(t))/(t-x)=0 . If f(x)=1 , then x is equal to :

Let f(x) be a differentiable function such that f'(x)+f(x)=4xe^(-x)sin2x and f(0)=0 Then the value of lim_(n rarr oo)sum_(k=1)^(n)f(k pi) is/are

Suppose that f is differentiable function with the property f(x+y)=f(x)+f(y) and lim_(x rarr0)(f(x))/(x)=100 ,then f'(2) is equal to N ,then the sum of digits of N is....