(8x-3)/(2x-1)-(4x2+1)/(4x2-1)=
[(8x-3)(2x+1)-(4x2+1)]/(4x2-1)=
(16x2+2x-3-4x2-1)/(4x2-1)=
(12x2+2x-4)/(4x2-1)=
2(6x2+x-2)/[(2x-1)(2x+1)]=
2(3x-1)(2x+1)/[(2x-1)(2x+1)]=
2(3x-1)/(2x-1) when x≠-½.
When x=½, the numerator=2(3/2-1)=1, while the denominator=0. Therefore the limit as x→½ is infinity.
When x=-½, the expression is 2(-3/2-1)/(-1-1)=(-5)/(-2)=5/2, therefore the limit as x→-½ is 5/2.