Digital SATMath: Advanced MathMedium
A scientist is modeling the concentration of a certain chemical, C(t), in micrograms per milliliter, in a solution over time t, in minutes. The function is C(t) = (3t^2 - 12) / (t^2 + t - 6). What is the value of C(t) as t approaches infinity?
- A3
- B1
- CUndefined
- D0
Show answer & explanationAnswer & explanation
Correct answer: A. 3
To find the limit of a rational function as t approaches infinity, compare the degrees of the numerator and denominator. Here, the degree of the numerator (t^2) is 2, and the degree of the denominator (t^2) is also 2. When the degrees are equal, the limit is the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, the limit is 3/1 = 3.
Why the other options are wrong
- B. This would be the case if the ratio of leading coefficients was 1.
- C. The limit exists and is a finite number.
- D. This would be the case if the degree of the numerator was less than the degree of the denominator.
Limits of Rational Functions at Infinity (Equal Degrees)
When finding the limit of a rational function as the variable approaches positive or negative infinity, if the degree of the numerator is equal to the degree of the denominator, the limit is the ratio of their leading coefficients.
- Degree of numerator = Degree of denominator.
- Limit = (leading coefficient of numerator) / (leading coefficient of denominator).
- This determines the horizontal asymptote of the function.
Memory trick: Degrees Equal? 'Ratio' the Leaders!