A certain type of bacteria decreases according to the model $P(t)=12,789 \mathrm{e}^{-0.45 t}$, where $P$ is the population of the bacteria after $t$ minutes. What will be the eventual population of this sample as time $t$ approaches infinity (round to the nearest whole bacteria)?
1,865
5,865
0
685
Answer
Final Answer: The eventual population of the bacteria as time t approaches infinity is \(\boxed{0}\).
Step 1 ：The question is asking for the limit of the function as t approaches infinity. In this case, the function is an exponential decay function, which means as t approaches infinity, the value of the function will approach 0. This is because the exponent of e is negative, which means as t increases, the value of e to that power decreases.
Step 2 ：Therefore, the eventual population of the bacteria will be 0.
Step 3 ：The result confirms my initial thought that as time t approaches infinity, the population of the bacteria will approach 0. This is because the function is an exponential decay function, and as t increases, the value of the function decreases.
Step 4 ：Final Answer: The eventual population of the bacteria as time t approaches infinity is \(\boxed{0}\).
