Digital SATMath: Advanced MathMedium
A chemist is analyzing a solution where the concentration of a certain chemical, C(t), in milligrams per liter, after t minutes is given by C(t) = (5t + 10) / (t + 2). What is the behavior of the concentration as time (t) approaches infinity?
- AThe concentration approaches 2 mg/L.
- BThe concentration approaches 0 mg/L.
- CThe concentration approaches infinity.
- DThe concentration approaches 5 mg/L.
Show answer & explanationAnswer & explanation
Correct answer: D. The concentration approaches 5 mg/L.
To find the behavior as t approaches infinity for a rational function, we look at the ratio of the leading coefficients of the numerator and denominator. C(t) = (5t + 10) / (t + 2). The leading coefficient in the numerator is 5, and in the denominator is 1. So, as t → ∞, C(t) → 5/1 = 5. (Alternatively, factor the numerator: C(t) = 5(t+2)/(t+2) = 5 for t ≠ -2. So the concentration is always 5 for t ≠ -2.)
Why the other options are wrong
- A. This might be a result of miscalculation or misinterpreting the coefficients.
- B. This would happen if the degree of the denominator was greater than the numerator.
- C. This would happen if the degree of the numerator was greater than the denominator.
Limits of Rational Functions at Infinity
To find the limit of a rational function as x approaches infinity (or negative infinity), compare the degrees of the numerator and denominator.
- If deg(num) < deg(den), limit is 0.
- If deg(num) > deg(den), limit is ±∞.
- If deg(num) = deg(den), limit is the ratio of leading coefficients.
- This limit represents the horizontal asymptote of the function.
Memory trick: Degree 'D'etermines 'D'irection at 'D'istant ends.