Question

If the function `f(x)=(4^sinx-1)^2/(xlog(1+2x))`  for x ≠ 0 is continuous at x = 0, find f (0).

Answer 1

f is continuous at x = 0.

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

`f(0)=lim_(x->0)(4^sinx-1)^2/(xlog(1+2x))`

`=lim_(x->0)((4^sinx-1)^2/x^2)/((xlog(1+2x))/x^2)`

`=lim_(x->0)(((4^sinx-1)/sinx)^2.sin^2x/x^2)/((2log(1+2x))/(2x))`

`=(lim_(x->0)(4^sinx-1)/sinx xx.lim_(x->0)sinx/x)^2/(2((lim_(x->0)log(1+2x))/(2x)))`

`f(0)=(log4)^2/2`