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, t hours after administration, is given by C(t) = (5t) / (t^2 + 1). What is the behavior of the drug concentration as time (t) approaches infinity?
- AC(t) approaches infinity
- BC(t) approaches 1
- CC(t) approaches 0
- DC(t) approaches 5
Show answer & explanationAnswer & explanation
Correct answer: C. C(t) approaches 0
To find the behavior as t approaches infinity for a rational function, compare the degrees of the numerator and denominator. Here, the degree of the numerator (5t) is 1, and the degree of the denominator (t^2 + 1) is 2. Since the degree of the denominator is greater than the degree of the numerator, the limit as t approaches infinity is 0.
Why the other options are wrong
- A. This would be the case if the degree of the numerator was greater than the denominator.
- B. This is incorrect, likely from misinterpreting coefficients or degrees.
- D. This would be the limit if the degrees were equal and the leading coefficients were 5 and 1, respectively.
Limits of Rational Functions at Infinity
For a rational function f(x) = P(x) / Q(x), the limit as x approaches infinity (or negative infinity) is determined by comparing the degrees of the numerator polynomial P(x) and the denominator polynomial Q(x).
- If deg(P) < deg(Q), limit is 0.
- If deg(P) = deg(Q), limit is the ratio of leading coefficients.
- If deg(P) > deg(Q), limit is ±infinity (or no horizontal asymptote).
Memory trick: Degree Rules: Bottom Big, Zero's the Prize!