`f(x)=((4^x-1)^3)/(sin(x/p)log(1+(x^2)/3)` is continuous at x=0 and` f(0)= 12(ln4)^3 `then p=

Options

A. 1
B. 2
C. 3
D. 4

Answer

D

Explanation

`because f(0)=RHL`
`therefore f(0)=lim_( x to 0^(+)) f(x)`
`rArr 12("In"4)^(3)=lim_(h to 0)((4^(h)-1)^(3))/(sin(h//p)*"In"(1+h^(2)//3))`
`rArr 12("In"4)^(3)=lim_(h to 0)(((4^(h)-1)/(h))^(3))/((sin(h//p))/((h//p))*("In"(1+h^(2)//3))/((h^(2)//3))*(1)/((3p)))`
`=(("In"4)^(3)*3p)/(1*1) " " therefore p =4`