lim x-->1/2 [(8x-3/2x-1) -(4x^ +1/4x^-1)]

Question

this type of Q asked in MT-CET

Answer 1

(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.