If `f(x)=sin^(-1)x` then prove that `lim_(x->1/2)f(3x-4x^3)=pi-3lim_(x->1/2)sin^(-1)x`

If f(x) = sin^(-1) x then prove that lim_(x rarr (1^(+))/(2)) f(3x -4x^(3)) = pi - 3 lim_(x rarr (1^(+))/(2)) sin^(-1) x

If f(x)=sin^(-1)x and lim_(xto1//2+)f(3x-4x^3)=a-3lim_(xto1//2+)f(x) , then [a] is equal to {where [] denotes G.I.F}

Prove that: 3sin^(-1)x=sin^(-1)(3x-4x^3), x in [-1/2,1/2]

Prove that: 3sin^(-1)x=sin^(-1)(3x-4x^3), x in [-1/2,1/2]

If (x^2+x−2)/(x+3) -1) f(x) then find the value of lim_(x->-1) f(x)

Which of the following is/are true? (a) lim_(x->oo)((2+x)^(40)(4+x)^5)/((2-x)^(45))=1 (b) lim_(x->0)(1-cos^3x)/(xsinxcosx)=3/2 (c) lim_(x->0)(ln(1+2x)-2"ln"(1+x))/(x^2)=-1 (d) lim_(x->oo)(cot^(-1)(sqrt(x+1)-sqrt(x)))/(sec^1((2x+1)/(x-1)^2)=1

If f(x) is monotonically increasing function for all x in R, such that f''(x)gt0andf^(-1)(x) exists, then prove that (f^(-1)(x_(1))+f^(-1)(x_(2))+f^(-1)(x_(3)))/(3)lt((f^(-1)(x_(1)+x_(2)+x_3))/(3))

Let f(x) be positive, continuous, and differentiable on the interval (a , b)a n d lim_(x->a^+)f(x)=1,lim_(x->b^-)f(x)=3^(1/4)dot If f^(prime)(x)geqf^3(x)+1/(f(x)), then the greatest value of b-a is (a) pi/(48) (b) pi/(36) (c) pi/(24) (d) pi/(12)

Statement 1: If f(x)=2/(pi) cot^(-1)((3x^(2)+1)/((x-1)(x-2))) , then lim_(xto1^(-))f(x)=0 and lim_(xto2^(-))f(x)=2 Statement 2: lim_(xtooo)cot^(-1)x=0 and lim_(xto -oo)cot^(-1)x=pi

If f(x)=cos^(-1)(4x^3-3x)and lim_(xto1/2+)f'(x)=a and lim_(xto1/2-)f'(x)=b then a + b+ 3 is equal to ____