Evaluate `underset(xrarr(pi)/(2))(lim) tan^(2)x[sqrt(2sin^(2)x+3sinx+4)-sqrt(sin^(2)x+6sinx+2)]`

Explanation

Rationalizing we get
`underset(xto pi//2)limx((2sin^(2)x+3sinx+4-sin^(2)x-6sinx-2))/(sqrt(2sin^(2)x+3sinx+4)+sqrt(sin^(2)x+6sinx+2))`
`=underset(xto pi//2)lim(sin^(2)x(sinx-1)(sinx-2))/((1-sin^(2)x)(sqrt(9)+sqrt(9)))`
`=underset(xto pi//2)lim(-sin^(2)x(sinx-2))/(6(1+sinx))`
`=(-1(1-2))/(6(1+1))=(1)/(12)`