ACT (Enhanced)MathematicsHard
A scientist is studying a chemical reaction where the concentration of a product, C(t), in moles per liter, after t minutes, is given by C(t) = 5 - 4e^(-0.1t). As time (t) approaches infinity, what value does the concentration C(t) approach?
- A0 moles/L
- B5 moles/L
- C1 moles/L
- D4 moles/L
Show answer & explanationAnswer & explanation
Correct answer: B. 5 moles/L
As t approaches infinity, the term e^(-0.1t) approaches 0 because the exponent becomes a very large negative number (e.g., e^(-large number) is very small). Therefore, 4e^(-0.1t) approaches 0. So, C(t) approaches 5 - 0 = 5.
Why the other options are wrong
- A. This would happen if the entire expression approached zero, which is not the case.
- C. Incorrect limit calculation.
- D. This would be the limit of 4e^(-0.1t) if it were 4, but it approaches 0. The 5 is the constant term.
Limit of Exponential Decay
For an exponential decay term e^(-kt) where k>0, as t approaches infinity, the term approaches 0. This often reveals a horizontal asymptote.
- e^(-kt) approaches 0 as t -> ∞.
- The limit of a sum/difference is the sum/difference of the limits.
- Reveals the long-term behavior or carrying capacity in models.
Memory trick: Negative 'N'umbers in the 'N'umerator make 'N'othing appear at infinity for 'e'.