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

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

• A. 3/2
• B. 2/3
• C. 1
• D. 0

Answer: (B)

When x becomes very large, the terms with the highest power dominate the behavior of a polynomial ratio. Here both the numerator and the denominator have degree 3, so factor out x^3 to see the limit clearly: (2x^3 + 5x^2 + 1)/(3x^3 + x) = (2 + 5/x + 1/x^3)/(3 + 1/x^2). As x → ∞, the terms with 1/x, 1/x^2, and 1/x^3 vanish, leaving 2/3. Therefore, the limit is 2/3.

When x becomes very large, the terms with the highest power dominate the behavior of a polynomial ratio. Here both the numerator and the denominator have degree 3, so factor out x^3 to see the limit clearly: (2x^3 + 5x^2 + 1)/(3x^3 + x) = (2 + 5/x + 1/x^3)/(3 + 1/x^2). As x → ∞, the terms with 1/x, 1/x^2, and 1/x^3 vanish, leaving 2/3. Therefore, the limit is 2/3.