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

Prove that sqrt((1-x)/(1+x)) is approximately equal to 1-x+x^(2)/2 when x is very small.

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

lim_(x rarr1)(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-1rarr0 d.L.H.L. not equal R.H.L.

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

If y=tan^(-1)[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))] then prove that (dy)/(dx)=(1)/(2sqrt(1-x^(2)))

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))

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))

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