`lim_(xto0) ((1-3^(x)-4^(x)+12^(x)))/(sqrt((2cosx+7))-3)`

Answer

`-12ln2xxln3`

Explanation

`underset(xto0)lim((1-3^(x)-4^(x)+12^(x)))/(sqrt((2cosx+7))-3)`
`=underset(xto0)lim((3^(x)-1)(4^(x)-1))/(sqrt((2cosx+7))-3)`
`=underset(xto0)lim((3^(x)-1)(4^(x)-1)(sqrt(2cosx+7))+3)/((2cosx+7-9))`
`=underset(xto0)lim(((3^(x)-1))/(x)xx((4^(x)-1))/(x)(sqrt((2cosx+7))+3))/((-2(1-cosx))/(x^(2)))`
`=(("1n "3)("1n "4)6)/(-2xx(1)/(2))=-6" 1n "3xx" 1n "4`
`=-12" 1n "2xx" 1n "3`