Digital SATMath: Advanced MathMedium
A pharmaceutical company is testing a new drug. The concentration of the drug in a patient's bloodstream, C(t), in mg/L, after t hours, is given by the function C(t) = (5t) / (t^2 + 1). What is the behavior of the drug concentration as t approaches infinity?
- AThe concentration approaches 5 mg/L.
- BThe concentration approaches 1 mg/L.
- CThe concentration increases indefinitely.
- DThe concentration approaches 0 mg/L.
Show answer & explanationAnswer & explanation
Correct answer: D. The concentration approaches 0 mg/L.
To determine the behavior as t approaches infinity, compare the degrees of the numerator and denominator. Since the degree of the denominator (2) is greater than the degree of the numerator (1), the limit as t approaches infinity is 0.
Why the other options are wrong
- A. This would be the limit if the degrees of the numerator and denominator were equal, and the ratio of leading coefficients was 5.
- B. This would be the limit if the degrees were equal and the ratio of leading coefficients was 1.
- C. This would occur if the degree of the numerator was greater than the degree of the denominator.
Limits of Rational Functions at Infinity
The limit of a rational function as the independent variable approaches infinity is determined by comparing the degrees of the numerator and denominator polynomials.
- If degree(numerator) < degree(denominator), limit is 0.
- If degree(numerator) = degree(denominator), limit is the ratio of leading coefficients.
- If degree(numerator) > degree(denominator), limit is ±infinity (no horizontal asymptote).
Memory trick: Compare the powers, infinity devours!