Compute lim_{x -> 3} (x^4 - 81) / (2x^2 - 5x - 3) A. 1 B. 53/12 C. 108/7 D. Limit does not exist

GATE 2019 · Engineering Mathematics · Limits · medium

Answer: 108/7

1. Factor numerator using difference of squares: x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9) = (x-3)(x+3)(x^2+9).
2. Factor denominator: Roots of 2x^2-5x-3=0: x = (5 +/- 7)/4, giving x=3 and x=-1/2. So 2x^2-5x-3 = 2(x-3)(x+1/2) = (x-3)(2x+1).
3. Cancel and evaluate: Cancel (x-3): limit = (3+3)(3^2+9)/(2*3+1) = (6)(18)/(7) = 108/7.