The value of `lim_(xto2) (2^(x)+2^(3-x)-6)/(sqrt(2^(-x))-2^(1-x))" is "`

Options

A. `|2x|gtsqrt(3)`
B. `|2x|ltsqrt(3)`
C. `|2x|gesqrt(3)`
D. `|2x|lesqrt(3)`

Answer

B

Explanation

`underset(xto2)lim(2^(x)+2^(3-x)-6)/(sqrt(2^(-x))-2^(1-x))`
`=underset(xto2)lim((2^(x))^(2)-6xx2^(x)+2^(3))/(sqrt(2^(x))-2)`[Multiplying `N^(r)` and `D^(r)` by `2^(x)`]
`=underset(xto2)lim((2^(x)-4)(2^(x)-2)"("sqrt(2^(x))+2")")/("("sqrt(2^(x))-2")""("sqrt(2^(x))+2")")`
`=underset(xto2)lim((2^(x)-4)(2^(x)-2)"("sqrt(2^(x))+2")")/("("2^(x)-4")")`
`underset(xto2)lim(2^(x)-2)(sqrt(2^(x))+2)=(2^(2)-2)(2+2)=8`