`lim_(x to 0) ((4^(x)-9^(x))/((4^(x)+9^(x)))) ` is equal to

Options

A. `log((2)/(3))`
B. `(1)/(2)log((3)/(2))`
C. `(1)/(2)log((3)/(2))`
D. `log((3)/(2))`

Answer

A

Explanation

`underset(x to 0)(lim)(4^(x)-9^(x))/(x(4^(x)+9^(x)))`
`=underset(x to 0)(lim((4^(x)log4-9^(x)log9)/(x(4^(x)log4+9^(x)log9))`
`+(4^(x)+9^(x))` (using L'Hospital's rule)
`=(log4-log9)/(2)=(2"log"(2)/(3))/(2)="log"(2)/(3)`.