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 mg/L, after t hours, is given by C(t) = (5t^2 + 3t + 2) / (2t^2 + 7). As time t approaches infinity, what value does the concentration C(t) approach?
- A0
- BUndefined
- C2/7
- D5/2
Show answer & explanationAnswer & explanation
Correct answer: D. 5/2
To find the limit of a rational function as t approaches infinity, compare the degrees of the numerator and denominator. Since the degrees are equal (both 2), the limit is the ratio of the leading coefficients. The ratio is 5/2.
Why the other options are wrong
- A. Incorrect. This would be the limit if the degree of the denominator was greater than the numerator.
- B. Incorrect. Rational functions typically approach a finite limit or infinity, not undefined, when comparing degrees.
- C. Incorrect. This is the ratio of the constant terms, not the leading coefficients.
Limits of Rational Functions at Infinity
For a rational function, the limit as the independent variable approaches infinity is determined by comparing the degrees of the numerator and denominator polynomials.
- If deg(num) < deg(den), limit is 0.
- If deg(num) = deg(den), limit is ratio of leading coefficients.
- If deg(num) > deg(den), limit is ±infinity.
Memory trick: Compare the powers, find the limit!