If `f(x)={{:(,((4^(x)-1)^(3))/(sin(x//4)log(1+x^(2)//3)),x ne 0),(,k,x=0):}` is a continous at x=0, then k=

Options

A. `12(log, 4)^(2)`
B. `96(log, 2)^(3)`
C. `(log, 4)^(3)`
D. none of these

Answer

B

Explanation

Since f(x) is a continous at x=0. Therefore,
`underset(x to 0)lim f(x)=f(0)`
`Rightarrow underset(x to 0)lim ((4^(x)-1)^(3))/("sin" (pi)/(4)log(1+x^(2)/(3)))=k`
`Rightarrow underset(x to 0)lim (12((4^(x)-1^(3))/(x)))/((("sin"(pi)/(4))/(pi//4))("log"(1+x^(2)//3)/(x^(2)//3)))=k`
`Rightarrow 12(log 4)^(3)=k Rightarrow k=96(log2)^(3)`