Prove `1/(sqrt(|x|-x)) ` exists when x < 0

If L=lim_(xto0) (1)/(x^(3))((1)/(sqrt(1+x))-(1+ax)/(1+bx)) exists,then

Prove each of the following tan^(-1) x=-pi +cot^(-1) 1/x=sin^(-1) (x)/(sqrt(1+x^(2)) =-cos^(-1) (1)/(sqrt(1+x^(2))" when "x lt 0

lim_(x->1)sqrt(1-cos2(x-1))/(x-1) a. exists and its equals sqrt(2) b. exists and its equals sqrt(-2) c.does not exist because x-1->0 d. L.H.L not equal R.H.L

Prove that cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sin x))/(sqrt(1 + sin x) - sqrt(1 - sin x))) = (x)/(2), x in (0, (pi)/(4))

The set of values of x for which the function f(x)=(1)/(x)+2^(sin^(-1)x)+(1)/(sqrt(x-2)) exists is

if y= (sin ^(-1)x)/(sqrt(1-x^2)) then prove that (1-x)^2) .d/dx=xy+1

If y=sqrt(x)+1/(sqrt(x)) , prove that 2x(dy)/(dx)=sqrt(x)-1/(sqrt(x))

Using intermediate value theorem, prove that there exists a number x such that x^(2005)+1/(1+sin^2x)=2005.

If y sqrt(x^(2)+1)= log (sqrt(x^(2)+1)-x) , prove that (x^(2)+1)(dy)/(dx) +xy+1=0 .

sqrt((3x)/(x+1))+sqrt((x+1)/(3x))=2," when "x ne 0and x ne -1