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(x)= lim_(n->oo)(sinx)^(2n)

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

Let f : R rarr R be a differentiable function at x = 0 satisfying f(0) = 0 and f'(0) = 1, then the value of lim_(x to 0) (1)/(x) . sum_(n=1)^(oo)(-1)^(n).f((x)/(n)) , is a. 0 b. -log2 c. 1 d. e

Let f : R rarr R be a differentiable function at x = 0 satisfying f(0) = 0 and f'(0) = 1, then the value of lim_(x to 0) (1)/(x) . sum_(n=1)^(oo)(-1)^(n).f((x)/(n)) , is

Let f : R rarr R be a differentiable function at x = 0 satisfying f(0) = 0 and f'(0) = 1, then the value of lim_(x to 0) (1)/(x) . sum_(n=1)^(oo)(-1)^(n).f((x)/(n)) , is

Let f : R to R be a real function. The function f is double differentiable. If there exists ninN and p in R 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 to R be a real function. The function f is double differentiable. If there exists ninN and p in R 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_(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