Which of the following functions is/are discontinuous at `x=1?` `f(x)=1/(1+2^(t a n x))` `g(x)=(lim)_(xvecoo)1/(1+n in s^2(pix))` `h(x)=2^-2^(((1/(1-x)))),x!=1a n dh(1)=1` `varphi(x)=(x-1)/(|x-1|+2(x-1)^2),x!=1a n dvarphi(1)=1`

If f(1)=1 and f'(1)=2 then lim_(xto1)((f(x))^(2)-1)/(x^(2)-1) is

Let f(x)=lim_(ntooo) (x)/(x^(2n)+1). Then

Discuss the continuity of f(x)=(lim)_(nvecoo)(x^(2n)-1)/(x^(2n)+1)

If L=("lim")_(nvecoo)(2x3^2x2^3x3^4..... x2^(n-1)x3^n)^(1/((n^(2+1))) , then the value of L^4 is

Column I: Function, Column II: Value of x for which both the functions in any option of column I are identical f(x)=tan^(-1)((2x)/(1-x^2)),g(x)=2tan^(-1)x , p. x in {-1,1} f(x)=(log)_(x2)25a n dg(x)=(log)_x5 , q. x in (-1,1) f(x)=sec^(-1)x+cos e c^(-1)x ,g(x)=sin^(-1)x+cos^(-1)x , r. x in (0,1)

Let the given function is differentiable at x = 1. f(x)={{:(lim_(nrarroo) (ax(x-1)(cot.(pix)/(4))^(n)+(px^(2)+2))/((cot,(pix)/(4))^(n)+1)",",x in(0,1)uu(1,2)),(0",",x=1):} Then the value of |a+q| is

Let f(x)=lim_(nrarroo) (tan^(-1)(tanx))/(1+(log_(x)x)^(n)),x ne(2n+1)(pi)/(2) then

Which of the following function is/are periodic? (a)f(x)={1,xi sr a t ion a l0,xi si r r a t ion a l (b)f(x)={x-[x]; 2n

If f(x)=log((1+x)/(1-x)),t h e n (a) f(x_1)f(x)=f(x_1+x_2) (b) f(x+2)-2f(x+1)+f(x)=0 (c) f(x)+f(x+1)=f(x^2+x) (d) f(x_1)+f(x_2)=f((x_1+x_2)/(1+x_1x_2))

Evaluate: ("lim")_(nvecoo)((a1 1/x+a2 1/x++a n1/x)/n)^(n x)