A chemical engineer is studying a reaction where the concentration of a reactant, C(t), in moles per liter, changes over time t, in seconds. The concentration is modeled by C(t) = (t^2 - 9) / (t^2 - t - 6). As time progresses, what value does the concentration approach?
- A1 mol/L
- B0 mol/L
- CUndefined
- D3 mol/L
Show answer & explanationAnswer & explanation
Correct answer: A. 1 mol/L
To find the value the concentration approaches as time progresses (t approaches infinity), we evaluate the limit of the rational function as t → ∞. Compare the degrees of the numerator and denominator. Both the numerator (t^2) and the denominator (t^2) have a degree of 2. When the degrees are equal, the limit is the ratio of their leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 1. Therefore, the limit is 1/1 = 1 mol/L.
Why the other options are wrong
- B. This would be the limit if the degree of the numerator was less than the degree of the denominator.
- C. The limit exists and is a finite number as t approaches infinity.
- D. This is a root of the numerator, not the limit at infinity.
Limits of Rational Functions at Infinity (Equal Degrees)
When evaluating the limit of a rational function as the independent variable approaches infinity, if the highest power in the numerator is equal to the highest power in 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 limit represents the horizontal asymptote of the function.
Memory trick: Same 'power' on top and bottom? Just 'share' the coefficients!