If n is a positive integer, prove that `underset(nrarrinfty)Lim(1+x+x^2+…….+x^(n-1))=1/(1-x),|x|<1`/

Prove that underset(xrarr0)lim (a^x-1)/x= log a .

Let f: R rarr R be a positive increasing function with underset(xrarrinfty)lim(f(3x))/(f(x))=1, then underset(xrarrinfty)lim(f(2x))/(f(x)) =

The coefficient of x^n , where n is any positive integer, in the expansion of (1+2x+3x^2 +... to oo)^(1//2) is:

Evaluate: underset(xrarr0)lim (sqrt(1+2x)-sqrt(1-2x))/(x)

underset(nrarrinfty)lim (1^2+2^2+3^2+………+n^2)/n^3 is equal to: