Compute the limit lim_{x->∞} (2x^2 - x)/(x^2 + 4x + 1).

Compute the limit lim_{x->∞} (2x^2 - x)/(x^2 + 4x + 1).

• A. 1
• B. 0
• C. ∞
• D. 2

Answer: (D)

As x grows large, the highest power terms dominate a rational function. Here both numerator and denominator are quadratic, so the limit equals the ratio of their leading coefficients. Divide by x^2 to see this clearly: (2 - 1/x) / (1 + 4/x + 1/x^2) → 2/1 = 2. Therefore, the limit is 2.

As x grows large, the highest power terms dominate a rational function. Here both numerator and denominator are quadratic, so the limit equals the ratio of their leading coefficients. Divide by x^2 to see this clearly: (2 - 1/x) / (1 + 4/x + 1/x^2) → 2/1 = 2. Therefore, the limit is 2.