`f(x)=(lim)_(n->oo)(sin(pix/2))^(2n)` for `x >0,x!=1`,and `f(1)=0.` Discuss the continuity at `x=1.`

Consider f(x) = lim_(x-oo)(x^n-sinx^n)/(x^n+sinx^n) for x>0,x!=1,f(1)=0 then

A function f(x) is defined by, f(x) = {{:(([x^(2)]-1)/(x^(2)-1)",","for",x^(2) ne 1),(0",","for",x^(2) = 1):} Discuss the continuity of f(x) at x = 1.

Let f(x)=lim_(n->oo)(log(2+x)-x^(2n)sinx)/(1+x^(2n)) . then

If f(x)=lim_(ntooo) n(x^(1//n)-1)," then for "xgt0, ygt0,f(xy) is equal to

If f(x)=(x+1)/(x-1)a n dg(x)=1/(x-2),t h e n discuss the continuity of f(x),g(x),a n dfog(x)dot

If f(x)={(cos^2x-sin^2x-1)/(sqrt(x^2+1)-1), x!=0,and f(x)=k , x=0" is continuous at "x=0,"find " k

If f (x) = lim _( n to oo) (n (x ^(1//n)-1)) for x gt 0, then which of the following is/are true?

Consider f(x) =x^(2)+ax+3 and g(x) =x+band F(x) = lim_( n to oo) (f(x)+x^(2n)g(x))/(1+x^(2n)) If F(x) is continuous at x=-1, then

Let f(x)=(lim)_(n rarr oo)sum_(r=0)^(n-1)x/((r x+1){(r+1)x+1}) . Then (A) f(x) is continuous but not differentiable at x=0 (B) f(x) is both continuous but not differentiable at x=0 (C) f(x) is neither continuous not differentiable at x=0 (D) f(x) is a periodic function.

If f(x)=lim_(n->oo)n(x^(1/n)-1),then for x >0,y >0,f(x y) is equal to : (a) f(x)f(y) (b) f(x)+f(y) (c) f(x)-f(y) (d) none of these