If `f: RvecR` be a differen tiable function atg `x=0` satisfying `f(0)=0a n df^(prime)(0)=1` , then the value of `(lim)_(xvec0)1/xsum_(n=1)^oo(-1)^nf(x/n)=` `e` b. `-log2` c. `0` d.`1`

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

If f: RvecR is a differentiable function such that f^(prime)(x)>2f(x)fora l lxRa n df(0)=1,t h e n : f(x) is decreasing in (0,oo) f^(prime)(x) e^(2x)in(0,oo)

If a function satisfies the relation f(x) f''(x)-f(x)f'(x)=(f'(x))^(2) AA x in R and f(0)=f'(0)=1, then The value of lim_(x to -oo) f(x) is

If f(n)=int_(0)^(2015)(e^(x))/(1+x^(n))dx , then find the value of lim_(nto oo)f(n)

If f(x) is differentiable and strictly increasing function,then the value of lim_(x rarr0)(f(x^(2))-f(x))/(f(x)-f(0)) is 1 (b) 0(c)-1 (d) 2

Suppose |[f'(x),f(x)],[f''(x),f'(x)]|=0 is continuously differentiable function with f^(prime)(x)!=0 and satisfies f(0)=1 and f'(0)=2 then (lim)_(x->0)(f(x)-1)/x is 1//2 b. 1 c. 2 d. 0

Le the be a real valued functions satisfying f(x+1) + f(x-1) = 2 f(x) for all x, y in R and f(0) = 0 , then for any n in N , f(n) =

If f: RvecR is a twice differentiable function such that f"(x)>0fora l lxR ,a n d f(1/2)=1/2,f(1)=1 then: f^(prime)(1)>1 (b) f^(prime)(1)lt=0 (c)1/2 ltf^(prime)(1)lt1 (d)="" 0ltf^(prime)(1)lt="1/2