If `f(x)=1+x+x^2+x^3+....oo` for `|x| < 1` then `f^-1(x)=`

If f(x)=x-x^2+x^3-x^4+....oo, |x| lt 1 then f'(x)=

Let f(x) = (x^2 - 4)/(x^2 + 4) , for |x| > 2 , then the function f: (-oo , -2] cup [2, oo) to (-1,1) is :

If f (x) = x-x ^(2) + x ^(3) - x ^(4) +….oo, |x| lt 1, then f '(x)=

If f(x)=(x)/(1+(In x)(ln x)...oo)AA x in[1,oo) then int_(1)^(2e)f(x)dx equals:

If f(x)=x-x^(2)+x^(3)-x^(4)+….oo when |x| lt 1 then f^(-1)(x)=

If f(x) = x/(1 + (In x)(ln x). ..oo) AA x in [1.oo) then int_1^(2e) f(x) dx equals:

If f(x)={x^3,if x^2 =1 then f(x) is differentiable at (a) (-oo,oo)-{1} (b) (-oo,oo)-{1,-1} (c) (-oo,oo)-{1,-1,0} (d) (-oo,oo)-{-1}

Let f(x) lt 0 AA x in (-oo, 0) and f (x) gt 0 ,AA x in (0,oo) also f (0)=0, Again f'(x) lt 0 ,AA x in (-oo, -1) and f '(x) gt 0, AA x in (-1,oo) also f '(-1)=0 given lim _(x to -oo) f (x)=0 and lim _(x to oo) f (x)=oo and function is twice differentiable. If f'(x) lt 0 AA x in (0,oo)and f'(0)=1 then number of solutions of equation f (x)=x ^(2) is : (a) 1 (b) 2 (c) 3 (d) 4

Statement 1: lim_ (x rarr oo) ((1 ^ (2)) / (x ^ (3)) + (2 ^ (2)) / (x ^ (3)) + (3 ^ (2)) / (x ^ (3)) + ...... + (x ^ (2)) / (x ^ (3))) = lim_ (x rarr oo) (1 ^ (2)) / (x ^ ( 3)) + lim_ (x rarr oo) (2 ^ (2)) / (x ^ (3)) + ...... + lim_ (x rarr a) (x ^ (2)) / (x ^ (3)) lim_ (x rarr a) (f_ (1) (x) + f_ (2) (x) + ... + f_ (n) (x)) = lim_ (x rarr a) f_ (1) (x) + lim_ (x rarr a) f (x) + ...... + lim_ (x rarr a) f_ (n) (x)