Evaluate the limit as x approaches infinity of (3x^2 + 2x + 1)/(2x^2 - 5).

Evaluate the limit as x approaches infinity of (3x^2 + 2x + 1)/(2x^2 - 5).

• A. 2/3
• B. 3/2
• C. 1
• D. 0

Answer: (B)

As x grows without bound, the terms with the highest power of x dominate both the numerator and the denominator. Since both have x^2 as the leading term, the limit equals the ratio of their leading coefficients: 3 in the numerator and 2 in the denominator. Dividing every term by x^2 gives (3 + 2/x + 1/x^2) / (2 - 5/x^2). As x → ∞, the terms with 1/x and 1/x^2 vanish, leaving 3/2. So the limit is 3/2.

As x grows without bound, the terms with the highest power of x dominate both the numerator and the denominator. Since both have x^2 as the leading term, the limit equals the ratio of their leading coefficients: 3 in the numerator and 2 in the denominator. Dividing every term by x^2 gives (3 + 2/x + 1/x^2) / (2 - 5/x^2). As x → ∞, the terms with 1/x and 1/x^2 vanish, leaving 3/2. So the limit is 3/2.