Digital SATMath: Advanced MathHard
A scientist is modeling the concentration of a drug in a patient's bloodstream over time. The concentration C(t), in milligrams per liter, after t hours, is given by the function C(t) = (5t + 10) / (t + 2). What is the concentration of the drug as t approaches infinity?
- A10 mg/L
- B5 mg/L
- C2 mg/L
- D0 mg/L
Show answer & explanationAnswer & explanation
Correct answer: B. 5 mg/L
To find the concentration as t approaches infinity for a rational function, we look at the degrees of the numerator and denominator. Since the degree of the numerator (5t + 10, degree 1) is equal to the degree of the denominator (t + 2, degree 1), the limit as t approaches infinity is the ratio of the leading coefficients. The leading coefficient of the numerator is 5, and the leading coefficient of the denominator is 1. So, the limit is 5/1 = 5.
Why the other options are wrong
- A. This is the constant term in the numerator, not the limit at infinity.
- C. Incorrectly simplified or interpreted. This might come from (10/2) if only constants were considered.
- D. This would be the limit if the degree of the denominator was greater than the degree of the numerator.
Limits of Rational Functions at Infinity (Equal Degrees)
For a rational function where the degree of the numerator is equal to the degree of the denominator, the limit as x approaches positive or negative infinity is the ratio of the leading coefficients.
- Applies when degree(numerator) = degree(denominator).
- The limit is a horizontal asymptote.
- Represents the long-term behavior of the function.
Memory trick: Degrees are equal, coefficients prevail.