" If "f(x)={[3-[cot^(-1)((2x^(3)-3))/(x^(2))],quad x>0],[{x^(2)}cos(e^(1/x)),quad x<0]" is continuous at "x=0" then the value of "f(0)" is "([x]" and "{x}

Given f(x)={{:(3-[cot^(-1)((2x^3-3)/(x^2))], x >0) ,({x^2}cos(e^(1/x)) , x<0):} (where {} and [] denotes the fractional part and the integral part functions respectively). Then which of the following statements do/does not hold good?

Given f(x)={3-[cot^(-1)((2x^3-3)/(x^2))] for x >0 and {x^2}cos(e^(1/x)) for x<0 (where {} and [] denotes the fractional part and the integral part functions respectively). Then which of the following statements do/does not hold good?

If f(x)=cot^(-1)((3x-x^(3))/(1-3x^(2))) and g(x)=cos^(-1)((1-x^(2))/(1+x^(2))) then lim_(x rarr a)(f(x)-f(a))/(g(x)-g(a))

If f(x) =cot ^(-1) ((3x -x^(3))/(1-3x^(2))) and g (x) =cos ^(-1) ((1- x ^(2))/(1+x^(2))) then lim _(xtoa) (f(x) -f(a))/(g (x) -g(a)) is equal to-

If f(x)=tan^(-1)((3x-x^(3))/(1-3x^(2))) and phi(x)=cos^(-1)((1-x^(2))/(1+x^(2))) , then the value of lim_(x to a) (f(x)-f(a))/(phi(x)-phi(a))(0 lt a lt (1)/(2)) is -

If f (x) = cot ^(-1)((3x -x ^(3))/( 1- 3x ^(2)))and g (x) = cos ^(-1) ((1-x ^(2))/(1+x^(2))) then lim _(xtoa)(f(x) - f(a))/( g(x) -g (a)), 0 ltalt 1/2 is :

If f (x) = cot ^(-1)((3x -x ^(3))/( 1- 3x ^(2)))and g (x) = cos ^(-1) ((1-x ^(2))/(1+x^(2))) then lim _(xtoa)(f(x) - f(a))/( g(x) -g (a)), 0 lt 1/2 is :

If f(x)=cot^(-1) ((3x-x^3)/(1-3x^2)) and g(x)=cos^(-1)((1-x^2)/(1+x^2)) then lim_(x->a) (f(x)-f(a))/(g(x)-g(a))

If f(x)=cos^(-1)x+cos^(-1){(x)/(2)+(1)/(2)sqrt(3-3x^(2))} then

Let f(x)=cot^(-1)((x^(2)-x+1)/(2x-3x^(2))+(x^(2)-x+1)/(3-2x)) and if f((3)/(2))+f((5)/(7))+f((2)/(3))+f((7)/(5))=k pi then k is