Sketch a possible graph of a function f that satisfies the following conditions :
1. `f'(x) gt 0` on `(-oo,1), f'(x) lt ` on `(1,oo)`
2. `f'(x) gt 0` on `(-oo,-2)` and `(2,oo), f'(x) lt 0` on `(-2, 2)`
3. `{:(" "lim),(x rarr -oo):} f(x)=-2, {:(" "lim),(x rarr oo):} f(x)=0`

lim_(x rarr oo) (1+f(x))^(1/f(x))

Let f(x)=lim_(n rarr oo)(2x^(2n)sin\ 1/x+x)/(1+x^(2n)) then find (a) lim_(x rarr oo) x f(x) (b) lim_(x rarr 1) f(x)

If f(x)=(x^(2))/(1+x^(2)), prove that lim_(x rarr oo)f(x)=1

Consider the function y=f(x) satisfying the condition f(x+(1)/(x))=x^(2)+1/x^(2)(x!=0) Then the domain of f(x) is R domain of f(x) is R-(-2,2) range of f(x) is [-2,oo] range of f(x) is (2,oo)

A function f(x) having the following properties, (i) f(x) is continuous except at x=3 (ii) f(x) is differentiable except at x=-2 and x=3 (iii) f(0) =0 lim_(x to 3) f(x) to - oo lim_(x to oo) f(x) =3 , lim_(x to oo) f(x)=0 (iv) f'(x) gt 0 AA in (-oo, -2) uu (3,oo) " and " f'(x) le 0 AA x in (-2,3) (v) f''(x) gt 0 AA x in (-oo,-2) uu (-2,0)" and "f''(x) lt 0 AA x in (0,3) uu(3,oo) Then answer the following questions Find the Maximum possible number of solutions of f(x)=|x|

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

f : [0, oo) rArr [0, oo), f(x) = (5x)/(5+x) is

The function f : (0, oo) rarr [0, oo), f(x) = (x)/(1+x) is

f (x) = (x + 2) / (x + 1) rArr f (-2) = (- 2 + 2) / (- 2 + 1) = 0x rarr-oo

If f : ( 0, oo) rarr ( 0, oo) and f ( x ) = ( x )/( 1+ x ) , then f is