The function `f: R-{0} -> R` given by `f(x)=1/x-2/[e^(2x)-1]` can be made continuous at x=0 by defining f(0) as

Explanation

`f(0) = lim_(x-> o-) f(x) = lim_(x->0+) f(x) = lim_(x->0) f(x)`

`f(0)= lim_(x->0) f(x) = lim_(x->0) [1/x - 2/(e^(2x) - 1)]`

`= lim_(x->0) (e^(2x)-1-2x)/(x(e^(2x) -1))`

`= lim_(x->0) (1 + 2x + (2x)^2/(2!) + ..... -1-2x)/(x(1 + 2x + (2x)^2/(2!) +.....-1))`

`= lim_(x->0) ((4x^2)/(2!) + (8x^3)/(3!) + ....)/(x^2(2 + (4x)/(2!) + (8x^2)/(3!)+ ....)`

`f(0) = (4/(2!))/2 = 4/(2 xx2)`

`f(0) = 1`

option D is correct