Determine lim_{x->∞} (4x^2 + 7x + 1)/(x^2 - x + 3).

Determine lim_{x->∞} (4x^2 + 7x + 1)/(x^2 - x + 3).

• A. 4
• B. 0
• C. ∞
• D. -∞

Answer: (A)

When evaluating a limit of a rational function as x grows without bound, the leading terms drive the behavior. Here both the numerator and the denominator are degree 2 polynomials, so the limit equals the ratio of their leading coefficients. Factor x^2 to see this clearly: (4 + 7/x + 1/x^2) / (1 - 1/x + 3/x^2). As x → ∞, the terms with 1/x and 1/x^2 vanish, leaving 4/1, which equals 4. So the limit is 4. The other options would arise if the degrees were different (0 if the numerator’s degree were lower, or ∞/−∞ if it were higher), which isn’t the case here.

When evaluating a limit of a rational function as x grows without bound, the leading terms drive the behavior. Here both the numerator and the denominator are degree 2 polynomials, so the limit equals the ratio of their leading coefficients.

Factor x^2 to see this clearly: (4 + 7/x + 1/x^2) / (1 - 1/x + 3/x^2). As x → ∞, the terms with 1/x and 1/x^2 vanish, leaving 4/1, which equals 4.

So the limit is 4. The other options would arise if the degrees were different (0 if the numerator’s degree were lower, or ∞/−∞ if it were higher), which isn’t the case here.