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

If f(x)=(x^(2))/(1+x^(2)), prove that lim_(x rarr oo)f(x)=1

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

Let f(x)=lim_(n->oo)(2x^(2n)sin(1/x)+x)/(1+x^(2n)) then find (a) lim_(x->oo) xf(x) (b) lim_(x->1) f(x) (c) lim_(x->0) f(x) (d) lim_(x->-oo) f(x)

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))

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

If f(x)={(3x-1),x>=1,(2x+3),x =2} then lim_(x rarr2)f(g(x))=

lim_(x-gt0)(cos2x-cos3x)/(cos4x-1)

lim_(x rarr0)(sin^(-1)x+3x)/(tan x+2sin((1)/(2)sin^(-1)x)[3-4sin^(2)((1)/(2)sin^(-1)x)])=