If `f (x)=underset(nrarroo)"lim"(1)/(1+x^(2n))` find f (x) for all real x .

If f(x)=underset(nrarroo)"lim"(1)/(1+nsin^(2)pix) , find f (x) for all real values of x .

Given , f(x)=x^(2)+ax+3,g(x)=x+band f (x)=underset(nrarroo)(lim)(f(x)+x^(2n)g(x))/(1+x^(2n)) If F (x) is continuous at x=+-1 , then g (x) = f (x) has -

Given , f(x)=x^(2)+ax+3,g(x)=x+band f (x)=underset(nrarroo)(lim)(f(x)+x^(2n)g(x))/(1+x^(2n)) If F (x) is continuous at x=-1 , then -

Given , f(x)=x^(2)+ax+3,g(x)=x+band f (x)=underset(nrarroo)(lim)(f(x)+x^(2n)g(x))/(1+x^(2n)) If F (x) is continuous at x = 1 , then -

Evaluate : underset(nrarroo)"lim"(1+sqrt(n))/(1-sqrt(n))

Statement -I : underset(xrarroo)"lim"(1^(2)/(x^(3))+(2^(2))/(x^(3))+3^(2)/(x^(3))+......+x^(2)/(x^(3)))= underset(xrarroo)"lim"(1^(2))/(x^(3))+underset(xrarroo)"lim"(2^(2))/(x^(3))+....+underset(xrarroo)"lim"(x^(2))/(x^(3))=0 Statement - II : underset(xrarra)"lim"{f_(1)(x)+f_(2)(x)+....+f_(n)(x)}=underset(xrarra)"lim"f_(1)(x)+underset(xrarra)"lim"f_(2)(x)+...+underset(xrarra)"lim"f_(n)(x)"

underset(xrarr1)"lim"(x+x^(2)+x^(3)+.....+x^(n)-n)/(x-1) =

If f(x) is a differentiable function in the interval (0, oo) such that f(1)=1 and underset(t to x)lim(t^(2)f(x)-x^(2)f(t))/(t-x)=1 , for each x gt 0 then f((3)/(2)) is equal to

Evaluate : underset(hrarr0)"lim"(f(1+h)-f(1))/(2h) , " where" f(x) =(1)/(x) .

Let f(x) be a polynomial of degree four having extreme values at x = 1 and x = 2. If underset(x to 0)lim[1+(f(x))/(x^(2))]=3 , then f(2) is equal to-