Evaluate lim_{x→∞} (4x^2 + 2x + 1)/(3x^2 - x + 5).

Evaluate lim_{x→∞} (4x^2 + 2x + 1)/(3x^2 - x + 5).

• A. 1
• B. 4/3
• C. 0
• D. ∞

Answer: (B)

When x grows large, the highest-degree terms in the numerator and denominator dominate the expression. So you can focus on the leading coefficients by factoring out x^2 from both parts: (4x^2 + 2x + 1) / (3x^2 − x + 5) = (4 + 2/x + 1/x^2) / (3 − 1/x + 5/x^2). As x approaches infinity, the terms with 1/x and 1/x^2 go to zero, leaving 4/3. Therefore, the limit is 4/3. This matches the chosen result because, with equal highest powers in top and bottom, the limit is the ratio of the leading coefficients. The other options would require different relationships between the degrees or leading coefficients (for instance, a limit of 0 if the top degree were lower, or ∞ if the top degree were higher).

When x grows large, the highest-degree terms in the numerator and denominator dominate the expression. So you can focus on the leading coefficients by factoring out x^2 from both parts:

(4x^2 + 2x + 1) / (3x^2 − x + 5) = (4 + 2/x + 1/x^2) / (3 − 1/x + 5/x^2).

As x approaches infinity, the terms with 1/x and 1/x^2 go to zero, leaving 4/3. Therefore, the limit is 4/3.

This matches the chosen result because, with equal highest powers in top and bottom, the limit is the ratio of the leading coefficients. The other options would require different relationships between the degrees or leading coefficients (for instance, a limit of 0 if the top degree were lower, or ∞ if the top degree were higher).