`(lim)_(xvec0^+)1/(xsqrt(x))("a t a n"^(-1)(sqrt(x))/a-btan^(-1)(sqrt(x))/b)` has the value equal to `(a-b)/3` b. 0 c. `((a^2-b^2))/(6a^2b^2)` d.`` `(a^2-b^2)/(3a^2b^2)`

If 3tan^(-1)(1/(2+sqrt(3)))-tan^(-1)(1/x)=tan^(-1)(1/3), then x is equal to 1 (b) 2 (c) 3 (d) sqrt(2)

If int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b ,t h e n a is e q u a l 1/3 (b) 2/3 (c) -1/3 (d0 -2/3

If af(x+1)+bf(1/(x+1))=x ,x!=-1,a!=b ,t h e nf(2) is equal to (a) (2a)/(2(a^2-b^2)) (b) a/(a^2-b^2) (c) (a+2b)/(a^2-b^2) (d) none of these

lim_(x->oo)[sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)] is equal to (a)0 (b) 1/2 (c) log 2 (d) e^4

("lim")_(ntooo)1/(1+nsin^2n x) is equal to (a) -1 (b) 0 (c) 1 (d) oo

If x = (sqrt(a+2b) + sqrt(a-2b))/(sqrt(a+2b) -sqrt(a-2b)) then prove that bx^2 -ax + b =0

"If "y=(2)/(sqrt(a^(2)-b^(2))){tan^(-1)(sqrt((a-b)/(a+b))tan""(x)/(2))}," then show that "(d^(2)y)/(dx^(2))=(b sin x)/((a+b cos x)^(2)).

If tan^(-1)x+2cot^(-1)x=(2pi)/3, then x , is equal to (a) (sqrt(3)-1)/(sqrt(3)+1) (b) 3 (c) sqrt(3) (d) sqrt(2)

If a = (sqrt5 + 1)/(sqrt5 + 1) and b = (sqrt5 -1)/(sqrt5 + 1) , then find the value of (a) (a^(2) + ab + b^(2))/(a^(2) - ab + b^(2)) (b) ((a -b)^(3))/((a + b)^(3)) (c) (3a^(2) + 5ab + b^(2))/(3a^(2) - 5ab + b^(2)) (d) (a^(3) + b^(3))/(a^(3) - b^(3))